diff --git a/changelog b/changelog index 01b91c6..812a22f 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,9 @@ +20071228 tpd src/hyper/bookvol11 add standards compliance for gamma +20071228 tpd src/hyper/bitmaps/gammacomplexinverse.png added +20071228 tpd src/hyper/bitmaps/gammacomplex.png added +20071228 tpd src/hyper/bitmaps/gammareal3.png added +20071228 tpd src/hyper/bitmaps/psi.png added +20071228 tpd src/hyper/bitmaps/loggamma.png added 20071225 sxw src/interp/bookvol5 fix top-level typo 20071218 acr src/algebra/axserver.spad fix lastType output re: errors 20071217 tpd src/algebra/variable.spad ignore regression test gensym (7041) diff --git a/src/hyper/bitmaps/gammacomplex.png b/src/hyper/bitmaps/gammacomplex.png new file mode 100644 index 0000000..cac4afd Binary files /dev/null and b/src/hyper/bitmaps/gammacomplex.png differ diff --git a/src/hyper/bitmaps/gammacomplexinverse.png b/src/hyper/bitmaps/gammacomplexinverse.png new file mode 100644 index 0000000..2ba6b6e Binary files /dev/null and b/src/hyper/bitmaps/gammacomplexinverse.png differ diff --git a/src/hyper/bitmaps/gammareal3.png b/src/hyper/bitmaps/gammareal3.png new file mode 100644 index 0000000..55be8e5 Binary files /dev/null and b/src/hyper/bitmaps/gammareal3.png differ diff --git a/src/hyper/bitmaps/loggamma.png b/src/hyper/bitmaps/loggamma.png new file mode 100644 index 0000000..bff7e62 Binary files /dev/null and b/src/hyper/bitmaps/loggamma.png differ diff --git a/src/hyper/bitmaps/psi.png b/src/hyper/bitmaps/psi.png new file mode 100644 index 0000000..275035c Binary files /dev/null and b/src/hyper/bitmaps/psi.png differ diff --git a/src/hyper/bookvol11.pamphlet b/src/hyper/bookvol11.pamphlet index 295ffe0..bd004ea 100644 --- a/src/hyper/bookvol11.pamphlet +++ b/src/hyper/bookvol11.pamphlet @@ -414,6 +414,32 @@ PAGES=rootpage.xhtml \ aldorusersguidepage.xhtml \ foundationlibrarydocpage.xhtml \ topicspage.xhtml \ + cats.xhtml \ + dlmf.xhtml \ + dlmfapproximations.xhtml \ + dlmfasymptoticexpansions.xhtml \ + dlmfbarnesgfunction.xhtml \ + dlmfbetafunction.xhtml \ + dlmfcontinuedfractions.xhtml \ + dlmfdefinitions.xhtml \ + dlmffunctionrelations.xhtml \ + dlmfgraphics.xhtml \ + dlmfinequalities.xhtml \ + dlmfinfiniteproducts.xhtml \ + dlmfintegrals.xhtml \ + dlmfintegralrepresentations.xhtml \ + dlmfmathematicalapplications.xhtml \ + dlmfmethodsofcomputation.xhtml \ + dlmfmultidimensionalintegral.xhtml \ + dlmfnotation.xhtml \ + dlmfphysicalapplications.xhtml \ + dlmfpolygammafunctions.xhtml \ + dlmfqgammaandbetafunctions.xhtml \ + dlmfseriesexpansions.xhtml \ + dlmfsums.xhtml \ + dlmfsoftware.xhtml \ + dlmfspecialvaluesandextrema.xhtml \ + dlmftables.xhtml \ uglangpage.xhtml \ examplesexposedpage.xhtml \ ugsyscmdpage.xhtml \ @@ -878,7 +904,8 @@ the javascript can be added easily. <>= + xmlns:xlink="http://www.w3.org/1999/xlink" + xmlns:m="http://www.w3.org/1998/Math/MathML"> Axiom Documentation @@ -1172,6 +1199,350 @@ is currently ignored. Name + Α + 913 + 00391 + &Alpha; + greek capital letter alpha + + + Β + 914 + 00392 + &Beta; + greek capital letter beta + + + Γ + 915 + 00393 + &Gamma; + greek capital letter gamma + + + Δ + 916 + 00394 + &Delta; + greek capital letter delta + + + Ε + 917 + 00395 + &Epsilon; + greek capital letter epsilon + + + Ζ + 918 + 00396 + &Zeta; + greek capital letter zeta + + + Η + 919 + 00397 + &Eta; + greek capital letter eta + + + Θ + 920 + 00398 + &Theta; + greek capital letter theta + + + Ι + 921 + 00399 + &Iota; + greek capital letter iota + + + Κ + 922 + 0039A + &Kappa; + greek capital letter kappa + + + Λ + 923 + 0039B + &Lambda; + greek capital letter lambda + + + Μ + 924 + 0039C + &Mu; + greek capital letter mu + + + Ν + 925 + 0039D + &Nu; + greek capital letter nu + + + Ξ + 926 + 0039E + &Xi; + greek capital letter xi + + + Ο + 927 + 0039F + &Omicron; + greek capital letter omicron + + + Π + 928 + 003A0 + &Pi; + greek capital letter pi + + + Ρ + 929 + 003A1 + &Rho; + greek capital letter rho + + + Σ + 931 + 003A3 + &Sigma; + greek capital letter sigma + + + Τ + 932 + 003A4 + &Tau; + greek capital letter tau + + + Υ + 933 + 003A5 + &Upsilon; + greek capital letter upsilon + + + Φ + 934 + 003A6 + &Phi; + greek capital letter phi + + + Χ + 935 + 003A7 + &Chi; + greek capital letter chi + + + Ψ + 936 + 003A8 + &Psi; + greek capital letter psi + + + Ω + 937 + 003A9 + &Omega; + greek capital letter omega + + + α + 945 + 003B1 + &alpha; + greek small letter alpha + + + β + 946 + 003B2 + &beta; + greek small letter beta + + + γ + 947 + 003B3 + &gamma; + greek small letter gamma + + + δ + 948 + 003B4 + &delta; + greek small letter delta + + + ε + 949 + 003B5 + &epsilon; + greek small letter epsilon + + + ζ + 950 + 003B6 + &zeta; + greek small letter zeta + + + η + 951 + 003B7 + &eta; + greek small letter eta + + + θ + 952 + 003B8 + &theta; + greek small letter theta + + + ι + 953 + 003B9 + &iota; + greek small letter iota + + + κ + 954 + 003BA + &kappa; + greek small letter kappa + + + λ + 955 + 003BB + &lambda; + greek small letter lambda + + + μ + 956 + 003BC + &mu; + greek small letter mu + + + ν + 957 + 003BD + &nu; + greek small letter nu + + + ξ + 958 + 003BE + &xi; + greek small letter xi + + + ο + 959 + 003BF + &omicron; + greek small letter omicron + + + π + 960 + 003C0 + &pi; + greek small letter pi + + + ρ + 961 + 003C1 + &rho; + greek small letter rho + + + ς + 962 + 003C2 + &sigmaf; + greek small letter final sigma + + + σ + 963 + 003C3 + &sigma; + greek small letter sigma + + + τ + 964 + 003C4 + &tau; + greek small letter tau + + + υ + 965 + 003C5 + &upsilon; + greek small letter upsilon + + + φ + 966 + 003C6 + &phi; + greek small letter phi + + + χ + 967 + 003C7 + &chi; + greek small letter chi + + + ψ + 968 + 003C8 + &psi; + greek small letter psi + + + ω + 969 + 003C9 + &omega; + greek small letter omega + + -------------------- + ¯ 175 000AF @@ -1179,6 +1550,20 @@ is currently ignored. macron + ± + 177 + 000B1 + &plusmn; + plus-or-minus sign + + + × + 215 + 000D7 + + multiplication sign + + è 232 000E8 @@ -1207,6 +1592,13 @@ is currently ignored. horizontal ellipsis + ⋯ + 8943 + 022EF + + midline horizontal ellipsis + + ′ 8242 02032 @@ -1228,6 +1620,83 @@ is currently ignored. invisible times + ℂ + 8450 + 02102 + + doube-struck captial c + + + ℍ + 8461 + 0210D + + double-struck captial h + + + ℑ + 8465 + 02111 + &image; + black-letter captial i + + + ℓ + 8467 + 02113 + + script small l + + + ℕ + 8469 + 02115 + + double-struck captial n + + + ℙ + 8473 + 02119 + + double-struck captial p + + + ℚ + 8474 + 0211A + + double-struck captial q + + + ℜ + 8476 + 0211C + &real; + black-letter captial r + + + ℝ + 8477 + 0211D + + double-struck captial r + + + ℤ + 8484 + 02124 + + double-struck captial z + + + ⅅ + 8517 + 02145 + + doube-struck captial d + + ⅆ 8518 02146 @@ -3051,6 +3520,7 @@ is currently ignored. <> @ + \subsection{aldorusersguidepage.xhtml} <>= <> @@ -5272,6 +5742,31 @@ the first k 20th powers. <> @ +\subsection{cats.xhtml} +<>= +<> + + +<> +
+ CATS -- Computer Algebra Test Suite +
+
+The Computer Algebra Test Suite is intended to show that Axiom conforms +to various published standards. Axiom implementations of these functions +are tested against reference publications. + +In order to show standards compliance we need to examine Axiom's behavior +against known good results. Where possible, these results are also tested +against other available computer algebra systems. + +The available test suites are: +
    +
  1. Gamma Function
    2. +
+<> +@ + \subsection{commandline.xhtml} <>= <> @@ -9256,6 +9751,20361 @@ the operations will have extra ones added at some stage. <> @ +\subsection{dlmf.xhtml} +<>= +<> + + +<> +
+ + Digital Library of Mathematical Functions +
+ The Gamma Function by R. A. Askey and R. Roy +
+
+

+The Gamma function is an extension of the factorial function to +real and complex numbers. For positive integers, + + Γ + + ( + n + ) + + = + + ( + n + - + 1 + ) + ! + +. +

+ +

+These pages explore Axiom's facilities for handling the Gamma function. +In particular we try to show that Axiom conforms to published standards. +

+ +<> +@ + +\subsection{dlmfapproximations.xhtml} +<>= +<> + + +<> +
+ + Digital Library of Mathematical Functions +
+ The Gamma Function -- Approximations +
+
+

Approximations

+
Contents
+
    +
  • Rational Approximations
    • +
  • Expansions in Chebyshev Series
    • +
  • Approximations in the Complex Plane
    • +
+ +

Rational Approximations

+ +

+ + Cody and Hillstrom(1967) + gives minimax rational approximations for + + + ln + + Γ + + ( + x + ) + + + + for the ranges + + + 0.5 + ≤ + x + ≤ + 1.5 + + , + + + + 1.5 + ≤ + x + ≤ + 4 + + , + + + 4 + ≤ + x + ≤ + 12 + + ; precision is variable. + + Hart et.al.(1968) + gives minimax polynomial and rational approximations to + + + + Γ + + ( + x + ) + + + and + + + ln + + Γ + + ( + x + ) + + + + in the intervals + + + + 0 + ≤ + x + ≤ + 1 + + , + + + + 8 + ≤ + x + ≤ + 1000 + + , + + + + 12 + ≤ + x + ≤ + 1000 + + ; precision is variable. + + + Cody et.al.(1973) + gives minimax rational approximations for + + + ψ + + ( + x + ) + + + for the ranges + + + + 0.5 + ≤ + x + ≤ + 3 + + and + + + + 3 + ≤ + x + < + ∞ + + ; precision is variable. +

+ +

For additional approximations see + + Hart et.al.(1968) + (Appendix B), + + Luke(1975) + (pp. 22–23), and + + Weniger(2003) + . +

+ +

Expansions in Chebyshev Series

+ +

+ + Luke(1969) + + gives the coefficients to 20D for the Chebyshev-series expansions of + + + Γ + + ( + + 1 + + + x + + ) + + + , + + + 1 + + Γ + + ( + + 1 + + + x + + ) + + + + , + + + + Γ + + ( + + x + + + 3 + + ) + + + , + + + + ln + + Γ + + ( + + x + + + 3 + + ) + + + + , + + + + ψ + + ( + + x + + + 3 + + ) + + + , and the first six derivatives of + + + + ψ + + ( + + x + + + 3 + + ) + + + for + + + + 0 + ≤ + x + ≤ + 1 + + . These coefficients are reproduced in + + Luke(1975) + . + + + Clenshaw(1962) + also gives 20D Chebyshev-series coefficients for + + + Γ + + ( + + 1 + + + x + + ) + + + and its reciprocal for + + + + 0 + ≤ + x + ≤ + 1 + + . See + + Luke(1975) + (pp. 22–23) for additional expansions. +

+ +

Approximations in the Complex Plane

+ +

Rational approximations for + + + + Γ + + ( + + z + + + 1 + + ) + + + + A + + ( + z + ) + + + + , where + + + + + A + + ( + z + ) + + + = + + + + ( + + 2 + π + + ) + + + 1 + 2 + + + + + ( + + z + + + c + + + + 1 + 2 + + + ) + + + z + + + 1 + + + + exp + + ( + + - + + ( + + z + + + c + + + + 1 + 2 + + + ) + + + ) + + + + + , and approximations for + + + + Γ + + ( + + z + + + 1 + + ) + + + based on the Padé approximants for two forms of the incomplete + gamma function are in + + Luke(1969) + . + + Luke(1975) + (pp. 13–16) provides explicit rational approximations for + + + + + ψ + + ( + z + ) + + + + + γ + + +

+<> +@ + +\subsection{dlmfasymptoticexpansions.xhtml} +<>= +<> + + +<> +
+ + Digital Library of Mathematical Functions +
+ The Gamma Function -- Asymptotic Expansions +
+
+

Asymptotic Expansions

+ +
Contents
+
    +
  • Poincaré-Type Expansions
    • +
  • Error Bounds and Exponential Improvement
    • +
  • Ratios
    • +
+ +

Poincaré-Type Expansions

+ +

As + + + z + → + ∞ + + in the sector + + + + | + + ph + + z + + | + + ≤ + + + π + - + δ + + + ( + + + < + π + + ) + + + + , +

+ + +
+ + + + + ln + + Γ + + ( + z + ) + + + + ∼ + + + + + ( + + z + - + + + 1 + 2 + + + + ) + + + ln + + z + + + - + z + + + + + + + 1 + 2 + + + + ln + + ( + + 2 + π + + ) + + + + + + + + ∑ + + k + = + 1 + + ∞ + + + + B + + 2 + k + + + + 2 + k + + ( + + + 2 + k + + - + 1 + + ) + + + z + + + 2 + k + + - + 1 + + + + + + + + + +
+ + +
+ + + + + ψ + + ( + z + ) + + + ∼ + + + ln + + z + + - + + 1 + + 2 + z + + + - + + + ∑ + + k + = + 1 + + ∞ + + + + B + + 2 + k + + + + 2 + k + + z + + 2 + k + + + + + + + + + +
+ +

For the Bernoulli numbers + + + B + + 2 + k + + + , + Also, +

+ + +
+ + + + + Γ + + ( + z + ) + + + ∼ + + + ⅇ + + - + z + + + + z + z + + + + ( + + + 2 + π + + z + + ) + + + 1 + 2 + + + + ( + + + ∑ + + k + = + 0 + + ∞ + + + + g + k + + + z + k + + + + ) + + + + + +
+ +
+ + + + + g + 0 + + = + 1 + + , + + + + + + + g + 1 + + = + + 1 + 12 + + + , + + + + + + + g + 2 + + = + + 1 + 288 + + + , + + + + + + + g + 3 + + = + + - + + 139 + 51840 + + + + , + + + + + + + g + 4 + + = + + - + + 571 + 24 88320 + + + + , + + + + + + + g + 5 + + = + + 1 63879 + 2090 18880 + + + , + + + + + + + g + 6 + + = + + 52 46819 + 7 52467 96800 + + + + +
+ +
+ + + + + g + k + + = + + + 2 + + + + ( + + + 1 + 2 + + + ) + + k + + + a + + 2 + k + + + + + , + + +
+ +

where + + + + a + 0 + + = + + + 1 + 2 + + + 2 + + + + , and +

+ +
+ + + + + + + a + 0 + + + a + k + + + + + + + + 1 + 2 + + + + a + 1 + + + a + + k + - + 1 + + + + + + + + + 1 + 3 + + + + a + 2 + + + a + + k + - + 2 + + + + + + … + + + + + + 1 + + k + + + 1 + + + + + a + k + + + a + 0 + + + + = + + + + 1 + k + + + + a + + k + - + 1 + + + + + + +
+ +
+ + + k + ≥ + 1 + + . +
+ +

+ + Wrench(1968) + gives exact values of + + + g + k + + up to + + + g + 20 + + . + + Spira(1971) + + corrects errors in Wrench's results and also supplies exact and 45D values of + + + g + k + + for + + + k + = + + 21 + , + 22 + , + … + , + 30 + + + . For an asymptotic expansion of + + + g + k + + as + + + k + → + ∞ + + see + Boyd(1994) + . +

+ +

With the same conditions +

+ +
+ + + + + Γ + + ( + + + a + z + + + + b + + ) + + + ∼ + + + + 2 + π + + + + ⅇ + + - + + a + z + + + + + + ( + + a + z + + ) + + + + + a + z + + + + b + + - + + ( + + 1 + 2 + + ) + + + + + + + +
+ +

where + + + a + + ( + + + > + 0 + + ) + + + and + + + b + + ( + + + ∈ + ℂ + + ) + + + are both fixed, and +

+ +
+ + + + + ln + + Γ + + ( + + z + + + h + + ) + + + + ∼ + + + + + ( + + + z + + + h + + - + + + 1 + 2 + + + + ) + + + ln + z + + + - + z + + + + + + + 1 + 2 + + + + ln + + ( + + 2 + π + + ) + + + + + + + + ∑ + + k + = + 2 + + ∞ + + + + + + ( + + - + 1 + + ) + + k + + + + B + k + + + ( + h + ) + + + + + k + + ( + + k + - + 1 + + ) + + + z + + k + - + 1 + + + + + + + + + +
+ +

where + + + h + + ( + + + ∈ + + [ + + 0 + , + 1 + + ] + + + ) + + + is fixed. +

+ +

Also as + + + y + → + + ± + ∞ + + + , +

+ +
+ + + + + | + + Γ + + ( + + x + + + + ⅈ + y + + + ) + + + | + + ∼ + + + + 2 + π + + + + + | + y + | + + + x + - + + ( + + 1 + 2 + + ) + + + + + ⅇ + + - + + + π + + | + y + | + + + 2 + + + + + + + +
+ +

uniformly for bounded real values of + + x + . +

+ +

Error Bounds and Exponential Improvement

+ +

If the sums in the expansions +(Equation 1) and +(Equation 2) are terminated at + + + k + = + + n + - + 1 + + + ( + + + k + ≥ + 0 + + ) and + + z + +is real and positive, then the remainder terms are bounded in magnitude by +the first neglected terms and have the same sign. If + + z + +is complex, then the remainder terms are bounded in magnitude by + + + + sec + + 2 + n + + + + ( + + + 1 + 2 + + + ph + + z + + + ) + + + for +(Equation 1), and + + + + sec + + + 2 + n + + + + 1 + + + + ( + + + 1 + 2 + + + ph + + z + + + ) + + + for +(Equation 2), times the first neglected terms.

+ +

For the remainder term in +(Equation 3) write +

+ +
+ + + + + Γ + + ( + z + ) + + + = + + + ⅇ + + - + z + + + + z + z + + + + ( + + + 2 + π + + z + + ) + + + 1 + 2 + + + + ( + + + + ∑ + + k + = + 0 + + + K + - + 1 + + + + + g + k + + + z + k + + + + + + + + R + K + + + ( + z + ) + + + + ) + + + + + +
+ +
+ + + K + = + + 1 + , + 2 + , + 3 + , + … + + + . +
+ +

Then +

+ +
+ + + + + | + + + R + K + + + ( + z + ) + + + | + + ≤ + + + + + ( + + 1 + + + + ζ + + ( + K + ) + + + + ) + + + Γ + + ( + K + ) + + + + + 2 + + + ( + + 2 + π + + ) + + + K + + + 1 + + + + + | + z + | + + K + + + + + ( + + 1 + + + + min + + ( + + + sec + + ( + + ph + + z + + ) + + + , + + 2 + + K + + + 1 + 2 + + + + + + ) + + + + ) + + + + + +
+ +
+ + + + | + + ph + z + + | + + ≤ + + + 1 + 2 + + π + + + +
+ + +

Ratios

+ +

If + + + a + + ( + + + ∈ + ℂ + + ) + + + and + + + b + + ( + + + ∈ + ℂ + + ) + + + are fixed as + + + z + → + ∞ + + in + + + + | + + ph + + z + + | + + ≤ + + + π + - + δ + + + ( + + + < + π + + ) + + + + , then +

+ +
+ + + + + + Γ + + ( + + z + + + a + + ) + + + + Γ + + ( + + z + + + b + + ) + + + + ∼ + + z + + a + - + b + + + + + +
+ +
+ + + + + + Γ + + ( + + z + + + a + + ) + + + + Γ + + ( + + z + + + b + + ) + + + + ∼ + + + z + + a + - + b + + + + + ∑ + + k + = + 0 + + ∞ + + + + + G + k + + + ( + + a + , + b + + ) + + + + z + k + + + + + + + +
+ +

Also, with the added condition + + + + ℜ + + ( + + b + - + a + + ) + + + > + 0 + + , +

+ +
+ + + + + + Γ + + ( + + z + + + a + + ) + + + + Γ + + ( + + z + + + b + + ) + + + + ∼ + + + + ( + + z + + + + + + a + + + b + + - + 1 + + 2 + + + ) + + + a + - + b + + + + + ∑ + + k + = + 0 + + ∞ + + + + + H + k + + + ( + + a + , + b + + ) + + + + + ( + + z + + + + + 1 + 2 + + + ( + + + a + + + b + + - + 1 + + ) + + + + ) + + + 2 + k + + + + + + + + +
+ +

Here +

+ +
+ + + + + + G + 0 + + + ( + + a + , + b + + ) + + + = + 1 + + , + + + + + + + + G + 1 + + + ( + + a + , + b + + ) + + + = + + + 1 + 2 + + + ( + + a + - + b + + ) + + + ( + + + a + + + b + + - + 1 + + ) + + + + , + + + + + + + + G + 2 + + + ( + + a + , + b + + ) + + + = + + + + 1 + 12 + + + + ( + + + + + + a + - + b + + + + + + 2 + + + + + ) + + + ( + + + 3 + + + ( + + + a + + + b + + - + 1 + + ) + + 2 + + + - + + ( + + + a + - + b + + + + 1 + + ) + + + ) + + + + + +
+ +
+ + + + + + H + 0 + + + ( + + a + , + b + + ) + + + = + 1 + + , + + + + + + + + H + 1 + + + ( + + a + , + b + + ) + + + = + + - + + + + 1 + 12 + + + + ( + + + + + + a + - + b + + + + + + 2 + + + + + ) + + + ( + + + a + - + b + + + + 1 + + ) + + + + + , + + + + + + + + H + 2 + + + ( + + a + , + b + + ) + + + = + + + + 1 + 240 + + + + ( + + + + + + a + - + b + + + + + + 4 + + + + + ) + + + ( + + + 2 + + ( + + + a + - + b + + + + 1 + + ) + + + + + + 5 + + + ( + + + a + - + b + + + + 1 + + ) + + 2 + + + + ) + + + + + +
+ +

In terms of generalized Bernoulli polynomials we have for + + + + k + = + + 0 + , + 1 + , + … + + + + +

+ +
+ + + + + + G + k + + + ( + + a + , + b + + ) + + + = + + + ( + + + + + a + - + b + + + + + + k + + + + ) + + + + B + k + + ( + + + a + - + b + + + + 1 + + ) + + + + ( + a + ) + + + + + + +
+ +
+ + + + + + H + k + + + ( + + a + , + b + + ) + + + = + + + ( + + + + + a + - + b + + + + + + + 2 + k + + + + + ) + + + + B + + 2 + k + + + ( + + + a + - + b + + + + 1 + + ) + + + + ( + + + + a + - + b + + + + 1 + + 2 + + ) + + + + + + +
+ +
+ + + + + + + Γ + + ( + + z + + + a + + ) + + + + Γ + + ( + + z + + + b + + ) + + + + + Γ + + ( + + z + + + c + + ) + + + + ∼ + + + ∑ + + k + = + 0 + + ∞ + + + + ( + + - + 1 + + ) + + k + + + + + + ( + + c + - + a + + ) + + k + + + + ( + + c + - + b + + ) + + k + + + + k + ! + + + + Γ + + ( + + + + + a + + + b + + - + c + + + + z + + - + k + + ) + + + + + + +
+<> +@ + +\subsection{dlmfbarnesgfunction.xhtml} +<>= +<> + + +<> +
+ + Digital Library of Mathematical Functions +
+ The Gamma Function -- Barnes G-Function (Double Gamma Function) +
+
+

Barnes + + G + -Function (Double Gamma Function) +

+ +
+ + + + + G + + ( + + z + + + 1 + + ) + + + = + + + Γ + + ( + z + ) + + + + + G + + ( + z + ) + + + + + , + + + + + + + G + + ( + 1 + ) + + + = + 1 + + , + + +
+ +
+ + + + + G + + ( + n + ) + + + = + + + + ( + + n + - + 2 + + ) + + ! + + + + ( + + n + - + 3 + + ) + + ! + + ⋯ + + 1 + ! + + + + + +
+ +
+ + + n + = + + 2 + , + 3 + , + … + + + +
+ +
+ + + + + G + + ( + + z + + + 1 + + ) + + + = + + + + + ( + + 2 + π + + ) + + + z + 2 + + + + exp + + ( + + + - + + + + 1 + 2 + + + + z + + + ( + + z + + + 1 + + ) + + + + - + + + + 1 + 2 + + + γ + + z + 2 + + + + ) + + + + × + + + ∏ + + k + = + 1 + + ∞ + + ( + + + + ( + + 1 + + + + z + k + + + ) + + k + + + exp + + ( + + + - + z + + + + + + z + 2 + + + 2 + k + + + + ) + + + + ) + + + + + +
+ +
+ + + + + Ln + + + G + + ( + + z + + + 1 + + ) + + + + = + + + + + + + 1 + 2 + + + + z + + + ln + + ( + + 2 + π + + ) + + + + - + + + + 1 + 2 + + + + z + + + ( + + z + + + 1 + + ) + + + + + + + z + + + Ln + + + Γ + + ( + + z + + + 1 + + ) + + + + + + - + + + ∫ + 0 + z + + + Ln + + + Γ + + ( + + t + + + 1 + + ) + + + + + + ⅆ + t + + + + + + +
+ +

The + + Ln + 's have their principal values on the positive real axis and are + continued via continuity. +

+ +

When + + + z + → + ∞ + + in + + + + | + + ph + + z + + | + + ≤ + + + π + - + δ + + + ( + + + < + π + + ) + + + + +

+ +
+ + + + + Ln + + + G + + ( + + z + + + 1 + + ) + + + + ∼ + + + + + + + 1 + 4 + + + + z + 2 + + + + + + z + + + Γ + + ( + + z + + + 1 + + ) + + + + + - + + + ( + + + + + 1 + 2 + + + + z + + + ( + + z + + + 1 + + ) + + + + + + + 1 + 12 + + + + ) + + + + Ln + + z + + + - + + ln + A + + + + + + + ∑ + + k + = + 1 + + ∞ + + + + B + + + 2 + k + + + + 2 + + + + 2 + k + + ( + + + 2 + k + + + + 1 + + ) + + + ( + + + 2 + k + + + + 2 + + ) + + + z + + 2 + k + + + + + + + + + +
+ +

see + + Ferreira and López(2001) +. This reference also provides bounds for the error term. Here + + + B + + + 2 + k + + + + 2 + + + is the Bernoulli number, and + + A + is Glaisher's constant, given by +

+ +
+ + + + A + = + + ⅇ + C + + = + + 1.28242 71291 00622 63687 + … + + + + +
+ +

where +

+ +
+ + + + C + = + + + lim + + n + → + ∞ + + + + ( + + + + + ∑ + + k + = + 1 + + n + + k + + + ln + + k + + + - + + + ( + + + + + 1 + 2 + + + + n + 2 + + + + + + + + 1 + 2 + + + n + + + + + + 1 + 12 + + + + ) + + + + ln + + n + + + + + + + + + 1 + 4 + + + + n + 2 + + + + ) + + + = + + + + γ + + + + ln + + ( + + 2 + π + + ) + + + + 12 + + - + + + + ζ + ′ + + + ( + 2 + ) + + + + 2 + + π + 2 + + + + + = + + + 1 + 12 + + - + + + ζ + ′ + + + ( + + - + 1 + + ) + + + + + + +
+ +

and + + + ζ + ′ + + is the derivative of the zeta function +

+ +

For Glaisher's constant see also + + Greene and Knuth(1982) + (p. 100). +

+<> +@ + +\subsection{dlmfbetafunction.xhtml} +<>= +<> + + +<> +
+ + Digital Library of Mathematical Functions +
+ The Gamma Function -- Beta Function +
+
+

Beta Function

+ +

In this section all fractional powers have their principal values, except +where noted otherwise. In the next 4 equations it is assumed + + + + ℜ + a + + > + 0 + + and + + + + ℜ + b + + > + 0 + + . +

+ +
Euler's Beta Integral
+ +
+ + + + + B + + ( + + a + , + b + + ) + + + = + + + ∫ + 0 + 1 + + + t + + a + - + 1 + + + + + ( + + 1 + - + t + + ) + + + b + - + 1 + + + + ⅆ + t + + + = + + + + Γ + + ( + a + ) + + + + Γ + + ( + b + ) + + + + + Γ + + ( + + a + + + b + + ) + + + + + + +
+ +
+ + + + + + ∫ + 0 + + π + 2 + + + + + sin + + + 2 + a + + - + 1 + + + θ + + + + cos + + + 2 + b + + - + 1 + + + θ + + + ⅆ + θ + + + = + + + + 1 + 2 + + + + B + + ( + + a + , + b + + ) + + + + + + +
+ +
+ + + + + + ∫ + 0 + ∞ + + + + + t + + a + - + 1 + + + + ⅆ + t + + + + + ( + + 1 + + + t + + ) + + + a + + + b + + + + + = + + B + + ( + + a + , + b + + ) + + + + + +
+ +
+ + + + + + ∫ + 0 + 1 + + + + + t + + a + - + 1 + + + + + ( + + 1 + - + t + + ) + + + b + - + 1 + + + + + + ( + + t + + + z + + ) + + + a + + + b + + + + + ⅆ + t + + + = + + + B + + ( + + a + , + b + + ) + + + + + ( + + 1 + + + z + + ) + + + - + a + + + + z + + - + b + + + + + + +
+ +

with + + + + | + + ph + z + + | + + < + π + + and the integration path along the real axis. +

+ +
+ + + + + + ∫ + 0 + + π + 2 + + + + + ( + + cos + + t + + ) + + + a + - + 1 + + + + cos + + ( + + b + t + + ) + + + + ⅆ + t + + + = + + + π + + 2 + a + + + + 1 + + a + + B + + ( + + + + 1 + 2 + + + ( + + a + + + b + + + 1 + + ) + + + , + + + 1 + 2 + + + ( + + + a + - + b + + + + 1 + + ) + + + + ) + + + + + + + + +
+ +
+ + + + ℜ + a + + > + 0 + + , +
+ +
+ + + + + + ∫ + 0 + π + + + + ( + + sin + + t + + ) + + + a + - + 1 + + + + ⅇ + + ⅈ + b + t + + + + ⅆ + t + + + = + + + π + + 2 + + a + - + 1 + + + + + + ⅇ + + + ⅈ + π + b + + 2 + + + + a + + B + + ( + + + + 1 + 2 + + + ( + + a + + + b + + + 1 + + ) + + + , + + + 1 + 2 + + + ( + + + a + - + b + + + + 1 + + ) + + + + ) + + + + + + + + +
+ +
+ + + + ℜ + a + + > + 0 + + , +
+ +
+ + + + + + ∫ + 0 + ∞ + + + + cosh + + ( + + 2 + b + t + + ) + + + + + ( + + cosh + + t + + ) + + + 2 + a + + + + + + ⅆ + t + + + = + + + 4 + + a + - + 1 + + + + B + + ( + + + a + + + b + + , + + a + - + b + + + ) + + + + + + +
+ +
+ + + + ℜ + a + + > + + | + + ℜ + b + + | + + + . +
+ +
+ + + + + + 1 + + 2 + π + + + + + ∫ + + - + ∞ + + ∞ + + + + ⅆ + t + + + + + ( + + w + + + + ⅈ + t + + + ) + + a + + + + ( + + z + - + + ⅈ + t + + + ) + + b + + + + + + = + + + + ( + + w + + + z + + ) + + + 1 + - + a + - + b + + + + + ( + + + a + + + b + + - + 1 + + ) + + + B + + ( + + a + , + b + + ) + + + + + + + +
+ +
+ + + + ℜ + + ( + + a + + + b + + ) + + + > + 1 + + , + + + + ℜ + w + + > + 0 + + , + + + + ℜ + z + + > + 0 + + +
+ +

The fractional powers have their principal values when + + + w + > + 0 + + and + + + z + > + 0 + + , and are continued via continuity. +

+ +
+ + + + + + 1 + + 2 + π + ⅈ + + + + + ∫ + + c + - + + ∞ + + ⅈ + + + + c + + + + ∞ + + ⅈ + + + + + t + + - + a + + + + + ( + + 1 + - + t + + ) + + + + - + 1 + + - + b + + + + ⅆ + t + + + + = + + 1 + + b + + B + + ( + + a + , + b + + ) + + + + + + + +
+ +
+ + + 0 + < + c + < + 1 + + , + + + + ℜ + + ( + + a + + + b + + ) + + + > + 0 + + +
+ +
+ + + + + + 1 + + 2 + π + ⅈ + + + + + ∫ + 0 + + ( + + 1 + + + + ) + + + + t + + a + - + 1 + + + + + ( + + t + - + 1 + + ) + + + b + - + 1 + + + + ⅆ + t + + + + = + + + + sin + + ( + + π + b + + ) + + + π + + + B + + ( + + a + , + b + + ) + + + + + + +
+ +
+ + + + ℜ + a + + > + 0 + + , +
+ +
+ +
+ +
+ + t + -plane. Contour for first loop integral for the beta function. +
+ +

In the next two equations the fractional powers are continuous on the + integration paths and take their principal values at the beginning. +

+ +
+ + + + + + 1 + + + ⅇ + + 2 + π + ⅈ + a + + + - + 1 + + + + + ∫ + ∞ + + ( + + 0 + + + + ) + + + + t + + a + - + 1 + + + + + ( + + 1 + + + t + + ) + + + + - + a + + - + b + + + + ⅆ + t + + + + = + + B + + ( + + a + , + b + + ) + + + + + +
+ +

when + + + + ℜ + b + + > + 0 + + , + + a + is not an integer and the contour cuts the real axis between + + + - + 1 + + and the origin. +

+ +
+ +
+ +
+ + t + -plane. Contour for second loop integral for the beta function. +
+ +
Pochhammer's Integral
+

When + + + + a + , + b + + ∈ + ℂ + + +

+ +
+ + + + + + ∫ + P + + ( + + + 1 + + + + , + + 0 + + + + , + + 1 + - + + , + + 0 + - + + + ) + + + + t + + a + - + 1 + + + + + ( + + 1 + - + t + + ) + + + b + - + 1 + + + + ⅆ + t + + + = + + - + + 4 + + ⅇ + + π + + ⅈ + + ( + + a + + + b + + ) + + + + + sin + + ( + + π + a + + ) + + + + sin + + ( + + π + b + + ) + + + + B + + ( + + a + , + b + + ) + + + + + + + +
+ +

where the contour starts from an arbitrary point + + P + in the interval + + + ( + + 0 + , + 1 + + ) + + ,circles + + 1 + and then + + 0 + in the positive sense, circles + + 1 + and then + + 0 + in the negative sense, and returns to + + P + . It can always be deformed into the contour shown here. +

+ +
+ +
+ +
+ + t + -plane. Contour for Pochhammer's integral. +
+<> +@ + +\subsection{dlmfcontinuedfractions.xhtml} +<>= +<> + + +<> +
+ + Digital Library of Mathematical Functions +
+ The Gamma Function -- Continued Fractions +
+
+

Continued Fractions

+ +

For + + + + ℜ + z + + > + 0 + + , +

+ +
+ + + + + + + ln + + Γ + + ( + z + ) + + + + + + z + + - + + + ( + + z + - + + + 1 + 2 + + + + ) + + + ln + + z + + + - + + + + 1 + 2 + + + + ln + + + ( + + 2 + π + + ) + + + + + = + + + + + a + 0 + + + + + + z + + + + + + + + + a + 1 + + + + + + z + + + + + + + + + a + 2 + + + + + + z + + + + + + + + + a + 3 + + + + + + z + + + + + + + + + a + 4 + + + + + + z + + + + + + + + a + 5 + + + z + + + + + + + + + + ⋯ + + + + +
+ +
+ + + + + a + 0 + + = + + 1 + 12 + + + , + + + + + + + a + 1 + + = + + 1 + 30 + + + , + + + + + + + a + 2 + + = + + 53 + 210 + + + , + + + + + + + a + 3 + + = + + 195 + 371 + + + , + + + + + + + a + 4 + + = + + 22999 + 22737 + + + , + + + + + + + a + 5 + + = + + 299 44523 + 197 33142 + + + , + + + + + + + a + 6 + + = + + 10 95352 41009 + 4 82642 75462 + + + + +
+ +

For rational values of + + + a + 7 + + to + + + a + 11 + + and 40S values of + + + a + 0 + + to + + + a + 40 + + , see + + Char(1980) +. Also see + + Jones and Thron(1980) +(pp. 348–350) and + + Lorentzen and Waadeland(1992) +(pp. 221–224) for further information. +

+<> +@ + +\subsection{dlmfdefinitions.xhtml} +<>= +<> + + +<> +
+ + Digital Library of Mathematical Functions +
+ The Gamma Function -- Definitions +
+
+

Definitions

+
Contents
+
    +
  • Gamma and Psi Functions
    • +
  • Euler's Constant
    • +
  • Pochhammer's Symbol
    • +
+

Gamma and Psi Functions

+
Euler's Integral
+ + + + + Γ + + ( + z + ) + + + = + + + ∫ + 0 + ∞ + + + ⅇ + + - + t + + + + t + + z + - + 1 + + + + ⅆ + t + + + + + + +
+ + + + ℜ + z + + > + 0 + + . +
+ +When + + + + ℜ + z + + ≤ + 0 + +, + + + + Γ + + ( + z + ) + + + is defined by analytic continuation. It is a meromorphic + function with no zeros, and with simple poles of residue + + + + + + ( + + - + 1 + + ) + + n + + + n + ! + + + at + + + + z + = + + - + n + + +. + + + + 1 + + Γ + + ( + z + ) + + + + is entire, with simple zeros at + + + + z + = + + - + n + + +. + +

+ + + + + ψ + + ( + z + ) + + + = + + + + Γ + ′ + + + ( + z + ) + + + + Γ + + ( + z + ) + + + + + + + +

+ + + z + ≠ + + 0 + , + + - + 1 + + , + + - + 2 + + , + … + + + +
+

+ +

+ + + ψ + + ( + z + ) + + + is meromorphic with simple poles of residue + + + - + 1 + + at + + + z + = + + - + n + + + . +

+ +

Euler's Constant

+ + + + γ + = + + + lim + + n + → + ∞ + + + + ( + + + 1 + + + + 1 + 2 + + + + + 1 + 3 + + + + … + + + + 1 + n + + + - + + ln + n + + + ) + + + = + + 0.57721 56649 01532 86060 + … + + + + + +

Pochhammer's Symbol

+
+ + + + + + ( + a + ) + + 0 + + = + 1 + + + + + + + + + + ( + a + ) + + n + + = + + a + + ( + + a + + + 1 + + ) + + + ( + + a + + + 2 + + ) + + ⋯ + + ( + + + a + + + n + + - + 1 + + ) + + + + + +
+ + + + + + + ( + a + ) + + n + + = + + + Γ + + ( + + a + + + n + + ) + + + + Γ + + ( + a + ) + + + + + + + +
+ + + a + ≠ + + + - + n + + , + + + - + n + + - + 1 + + , + + + - + n + + - + 2 + + , + … + + + +
+<> +@ + +\subsection{dlmffunctionrelations.xhtml} +<>= +<> + + +<> +
+ + Digital Library of Mathematical Functions +
+ The Gamma Function -- Function Relations +
+
+

Functional Relations

+
Contents
+
    +
  • Recurrence
    • +
  • Reflection
    • +
  • Multiplication
    • +
  • Bohr-Mollerup Theorem
    • +
+

Recurrence

+
+ + + + + Γ + + ( + + z + + + 1 + + ) + + + = + + z + + Γ + + ( + z + ) + + + + + + +
+ +
+ + + + + ψ + + ( + + z + + + 1 + + ) + + + = + + + ψ + + ( + z + ) + + + + + + 1 + z + + + + + +
+ + +

Reflection

+ +
+ + + + + + Γ + + ( + z + ) + + + + Γ + + ( + + 1 + - + z + + ) + + + + = + + π + + sin + + ( + + π + z + + ) + + + + + + +
+ +
+ + + z + ≠ + + 0 + , + + ŷ + 1 + + , + … + + + , +
+ +
+ + + + + + ψ + + ( + z + ) + + + - + + ψ + + ( + + 1 + - + z + + ) + + + + = + + - + + π + + tan + + ( + + π + z + + ) + + + + + + + +
+ +
+ + + z + ≠ + + 0 + , + + ŷ + 1 + + , + … + + + . +
+ +

Multiplication

+
+ + + + 2 + z + + ≠ + + 0 + , + + - + 1 + + , + + - + 2 + + , + … + + + , +
+ +
+ + + + + Γ + + ( + + 2 + z + + ) + + + = + + + π + + - + + 1 + 2 + + + + + 2 + + + 2 + z + + - + 1 + + + + Γ + + ( + z + ) + + + + Γ + + ( + + z + + + + + 1 + 2 + + + + ) + + + + + + +
+ +
+ + + + 3 + z + + ≠ + + 0 + , + + - + 1 + + , + + - + 2 + + , + … + + + , +
+ +
+ + + + + Γ + + ( + + 3 + z + + ) + + + = + + + + ( + + 2 + π + + ) + + + - + 1 + + + + 3 + + + 3 + z + + - + + ( + + 1 + 2 + + ) + + + + + Γ + + ( + z + ) + + + + Γ + + ( + + z + + + + + 1 + 3 + + + + ) + + + + Γ + + ( + + z + + + + + 2 + 3 + + + + ) + + + + + + +
+ +
+ + + + n + z + + ≠ + + 0 + , + + - + 1 + + , + + - + 2 + + , + … + + + , +
+ +
+ + + + + Γ + + ( + + n + z + + ) + + + = + + + + ( + + 2 + π + + ) + + + + ( + + 1 + - + n + + ) + + 2 + + + + n + + + n + z + + - + + ( + + 1 + 2 + + ) + + + + + + ∏ + + k + = + 0 + + + n + - + 1 + + + Γ + + ( + + z + + + + k + n + + + ) + + + + + + +
+ +
+ + + + + + ∏ + + k + = + 1 + + + n + - + 1 + + + Γ + + ( + + k + n + + ) + + + = + + + + ( + + 2 + π + + ) + + + + ( + + n + - + 1 + + ) + + 2 + + + + n + + - + + 1 + 2 + + + + + + + +
+ +
+ + + + + ψ + + ( + + 2 + z + + ) + + + = + + + + + 1 + 2 + + + + ( + + + ψ + + ( + z + ) + + + + + + ψ + + ( + + z + + + + + 1 + 2 + + + + ) + + + + ) + + + + + + ln + 2 + + + + + +
+ +
+ + + + + ψ + + ( + + n + z + + ) + + + = + + + + 1 + n + + + + ∑ + + k + = + 0 + + + n + - + 1 + + + ψ + + ( + + z + + + + k + n + + + ) + + + + + + + ln + n + + + + + +
+ + +

Bohr-Mollerup Theorem

+ +
+If a positive function + + + f + + ( + x + ) + + + on + + + ( + + 0 + , + ∞ + + ) + + satisfies + + + + f + + ( + + x + + + 1 + + ) + + + = + + x + + f + + ( + x + ) + + + + +, + + + + + f + + ( + 1 + ) + + + = + 1 + +, and + + + + ln + + f + + ( + x + ) + + + + is convex, then + + + + + f + + ( + x + ) + + + = + + Γ + + ( + x + ) + + + +. +<> +@ + +\subsection{dlmfgraphics.xhtml} +<>= +<> + + +<> +
+ + Digital Library of Mathematical Functions +
+ The Gamma Function -- Graphics +
+
+

Graphics

+
Contents
+
    +
  • Real Argument
    • +
  • The Psi Function
    • +
  • Complex Argument
    • +
+

Real Argument

+ +
+This graph shows the + + + Γ + + ( + x + ) + + + and + + + 1 + + Γ + + ( + x + ) + + + + . + +To create these two graphs in Axiom: +
+ -- Draw the first graph in a viewport
+ viewport1:=draw(Gamma(i), i=-4.2..4, adaptive==true, unit==[1.0,1.0])
+ -- Draw the second graph in a viewport
+ viewport2:=draw(1/Gamma(i), i=-4.2..4, adaptive==true, unit==[1.0,1.0])
+ -- Get the Gamma graph from the first viewport and layer it on top
+ putGraph(viewport2,getGraph(viewport1,1),2)
+ -- Remove the points and leave the lines
+ points(viewport2,1,"off")
+ points(viewport2,2,"off")
+ -- Show the combined graph
+ makeViewport2D(viewport2)
+
+ + +
+ + + ln + + Γ + + ( + x + ) + + + + . This function is convex on + + + ( + + 0 + , + ∞ + + ) + + ; +
+ compare + Functional Relations +

+You can construct this graph with the Axiom commands: +

+  -- draw the graph of log(Gamma) in a viewport
+  viewport1:=draw(log Gamma(i), i=0..8, adaptive==true, unit==[1.0,1.0])
+  -- turn off the points and leave the lines
+  points(viewport1,1,"off")
+
+

+
+ +

The Psi Function + + + ψ + + ( + x + ) + + + +

+ +

This function is a special case of the polygamma function. +In particular, + + + ψ + + ( + x + ) + + + is equal to polygamma(0,x). +

+
+
+ + +
+You can reconstruct this graph in Axiom by: +
+  -- first construct the psi function
+  psi(x)==polygamma(0,x)
+  -- draw the graph in a viewport
+  viewport:=draw(psi(y),y=-3.5..4,adaptive==true)
+  -- make the gradient obvious
+  scale(viewport,1,0.9,22.5)
+  -- and recenter the graph
+  translate(viewport,1,0,-0.02)
+  -- turn off the points and keep the line
+  points(viewport,1,"off")
+
+ +

Complex Argument

+ + +
+ + + + Γ + + ( + + x + + + + ⅈ + y + + + ) + + + + . +
+ +You can reconstruct this image in Axiom with: +
+  -- Set up the default viewpoint
+  viewPhiDefault(-%pi/4)
+  -- define the point set function
+  gam(x,y)== 
+    g:=Gamma complex(x,y) 
+    point [x,y,max(min(real g,4),-4), argument g] 
+  -- draw the image and remember the viewport
+  viewport:=draw(gam, -4..4,-3..3,var1Steps==100,var2Steps==100)
+  -- set the color mapping for the image
+  colorDef(viewport,blue(),blue())
+  -- and smoothly shade it
+  drawStyle(viewport,"smooth")
+
+ +
+ + + 1 + + + Γ + + ( + + x + + + + ⅈ + y + + + ) + + + + + +
+ +

+You can reproduce this image from Axiom with: +

+  -- Set up the default viewpoint
+  viewPhiDefault(-%pi/4)
+  -- Define the complex Gamma inverse function
+  gaminv(x,y)== 
+    g:=1/(Gamma complex(x,y)) 
+    point [x,y,max(min(real g,4),-4), argument g]
+  -- draw the 3D image and remember the viewport
+  viewport:=draw(gaminv, -4..4,-3..3,var1Steps==100,var2Steps==100)
+  -- make the image a uniform color
+  colorDef(viewport,blue(),blue())
+  -- and make it pretty
+  drawStyle(viewport,"smooth")
+
+

+ + +

+To get these exact images with the colored background you need +to use GIMP to set the background. The steps I used are: +

    +
  1. Save the image as a pixmap
    2. +
  2. Open the saved file in gimp
    3. +
  3. Dialogs->Colors->ColorPicker button
    4. +
  4. Eyedrop the color of the web page
    5. +
  5. Set the color as the foreground on the FG/BG page
    6. +
  6. Dialogs->Layers
    7. +
  7. Duplicate Layer
    8. +
  8. Layer->Stack->Select bottom layer
    9. +
  9. Edit->Fill with Foreground color
    10. +
  10. (on Layers panel)Select image
    11. +
  11. (on Layers panel) Mode->Darken Only
    12. +
+Note that you may have to use "lighten only" first before it will +allow you to choose "darken only". +

+ +<> +@ + +\subsection{dlmfinequalities.xhtml} +<>= +<> + + +<> +
+ + Digital Library of Mathematical Functions +
+ The Gamma Function -- Inequalities +
+
+

Inequalities

+
Contents
+
    +
  • Real Variables
    • +
  • Complex Variables
    • +
+ +

Real Variables

+

Throughout this subsection + + + x + > + 0 + +. +

+ +
+ + + + 1 + < + + + + ( + + 2 + π + + ) + + + - + + 1 + 2 + + + + + x + + + ( + + 1 + 2 + + ) + + - + x + + + + ⅇ + x + + + Γ + + ( + x + ) + + + + < + + ⅇ + + 1 + + ( + + 12 + x + + ) + + + + + + +
+ +
+ + + + + + 1 + + Γ + + ( + x + ) + + + + + + + 1 + + Γ + + ( + + 1 + x + + ) + + + + + ≤ + 2 + + + +
+ +
+ + + + + + 1 + + + ( + + Γ + + ( + x + ) + + + ) + + 2 + + + + + + 1 + + + ( + + Γ + + ( + + 1 + x + + ) + + + ) + + 2 + + + + ≤ + 2 + + + +
+ +
+ + + + + x + + 1 + - + s + + + < + + + Γ + + ( + + x + + + 1 + + ) + + + + Γ + + ( + + x + + + s + + ) + + + + < + + + ( + + x + + + 1 + + ) + + + 1 + - + s + + + + + +
+ +
+ + + 0 + < + s + < + 1 + + +
+ +
+ + + + + exp + + ( + + + ( + + 1 + - + s + + ) + + + ψ + + ( + + x + + + + s + + 1 + 2 + + + + ) + + + + ) + + + ≤ + + + Γ + + ( + + x + + + 1 + + ) + + + + Γ + + ( + + x + + + s + + ) + + + + ≤ + + exp + + ( + + + ( + + 1 + - + s + + ) + + + ψ + + ( + + x + + + + + + 1 + 2 + + + + ( + + s + + + 1 + + ) + + + + ) + + + + ) + + + + + +
+ +
+ + + 0 + < + s + < + 1 + + +
+ +

Complex Variables

+
+ + + + + | + + Γ + + ( + + x + + + + ⅈ + y + + + ) + + + | + + ≤ + + | + + Γ + + ( + x + ) + + + | + + + + +
+ +
+ + + + + | + + Γ + + ( + + x + + + + ⅈ + y + + + ) + + + | + + ≥ + + + + ( + + sech + + ( + + π + y + + ) + + + ) + + + 1 + 2 + + + + Γ + + ( + x + ) + + + + + + +
+ +
+ + + x + ≥ + + 1 + 2 + + + +
+ +

For + + + + b + - + a + + ≥ + 1 + + , + + + a + ≥ + 0 + + , and + + + z + = + + x + + + + ⅈ + y + + + + with + + + x + > + 0 + + , +

+ +
+ + + + + ∣ + + + Γ + + ( + + z + + + a + + ) + + + + Γ + + ( + + z + + + b + + ) + + + + ∣ + + ≤ + + 1 + + + | + z + | + + + b + - + a + + + + + + +
+ +

For + + + x + ≥ + 0 + + , +

+ +
+ + + + + | + + Γ + + ( + z + ) + + + | + + ≤ + + + + ( + + 2 + π + + ) + + + 1 + 2 + + + + + | + z + | + + + x + - + + ( + + 1 + 2 + + ) + + + + + ⅇ + + - + + + π + + | + y + | + + + 2 + + + + + exp + + ( + + + + 1 + 6 + + + + + | + z + | + + + - + 1 + + + + ) + + + + + + +
+<> +@ + +\subsection{dlmfinfiniteproducts.xhtml} +<>= +<> + + +<> +
+ + Digital Library of Mathematical Functions +
+ The Gamma Function -- Infinite Products +
+
+

Infinite Products

+ +
+ + + + + Γ + + ( + z + ) + + + = + + + lim + + k + → + ∞ + + + + + + k + ! + + + k + z + + + + z + + ( + + z + + + 1 + + ) + + ⋯ + + ( + + z + + + k + + ) + + + + + + + +
+ +
+ + + z + ≠ + + 0 + , + + - + 1 + + , + + - + 2 + + , + … + + + , +
+ +
+ + + + + 1 + + Γ + + ( + z + ) + + + + = + + z + + ⅇ + + γ + z + + + + + ∏ + + k + = + 1 + + ∞ + + + ( + + 1 + + + + z + k + + + ) + + + ⅇ + + - + + z + k + + + + + + + + +
+ +
+ + + + + + | + + + Γ + + ( + x + ) + + + + Γ + + ( + + x + + + + ⅈ + y + + + ) + + + + | + + 2 + + = + + + ∏ + + k + = + 0 + + ∞ + + ( + + 1 + + + + + y + 2 + + + + ( + + x + + + k + + ) + + 2 + + + + ) + + + , + + +
+ +
+ + + x + ≠ + + 0 + , + + - + 1 + + , + … + + + . +
+ +
+ + + + + + ∑ + + k + = + 1 + + m + + + a + k + + + = + + + ∑ + + k + = + 1 + + m + + + b + k + + + + + +
+ +

then +

+ +
+ + + + + + ∏ + + k + = + 0 + + ∞ + + + + + ( + + + a + 1 + + + + k + + ) + + + ( + + + a + 2 + + + + k + + ) + + ⋯ + + ( + + + a + m + + + + k + + ) + + + + + ( + + + b + 1 + + + + k + + ) + + + ( + + + b + 2 + + + + k + + ) + + ⋯ + + ( + + + b + m + + + + k + + ) + + + + + = + + + + Γ + + ( + + b + 1 + + ) + + + + Γ + + ( + + b + 2 + + ) + + + ⋯ + + Γ + + ( + + b + m + + ) + + + + + + Γ + + ( + + a + 1 + + ) + + + + Γ + + ( + + a + 2 + + ) + + + ⋯ + + Γ + + ( + + a + m + + ) + + + + + + + +
+ +

provided that none of the + + + b + k + + + is zero or a negative integer. +

+<> +@ + +\subsection{dlmfintegrals.xhtml} +<>= +<> + + +<> +
+ + Digital Library of Mathematical Functions +
+ The Gamma Function -- Integrals +
+
+

Integrals

+ + +
+ + + + + + 1 + + 2 + π + ⅈ + + + + + ∫ + + c + - + + ∞ + + ⅈ + + + + c + + + + ∞ + + ⅈ + + + + + Γ + + ( + + s + + + a + + ) + + + + Γ + + ( + + b + - + s + + ) + + + + z + + - + s + + + + ⅆ + s + + + + = + + + + Γ + + ( + + a + + + b + + ) + + + + z + a + + + + + ( + + 1 + + + z + + ) + + + a + + + b + + + + + + +
+ +
+ + + + ℜ + + ( + + a + + + b + + ) + + + > + 0 + + , + + + + - + + ℜ + a + + + < + c + < + + ℜ + b + + + , + + + + | + + ph + + z + + | + + < + π + + . +
+ +
+ + + + + + 1 + + 2 + π + + + + + ∫ + + - + ∞ + + ∞ + + + + | + + Γ + + ( + + a + + + + ⅈ + t + + + ) + + + | + + 2 + + + ⅇ + + + ( + + + 2 + b + + - + π + + ) + + t + + + + ⅆ + t + + + + = + + + Γ + + ( + + 2 + a + + ) + + + + + ( + + 2 + + sin + + b + + + ) + + + 2 + a + + + + + + +
+ +
+ + + a + > + 0 + + , + + + 0 + < + b + < + π + + . +
+ +
Barnes's Beta Integral
+ +
+ + + + + + 1 + + 2 + π + + + + + ∫ + + - + ∞ + + ∞ + + + Γ + + ( + + a + + + + ⅈ + t + + + ) + + + + Γ + + ( + + b + + + + ⅈ + t + + + ) + + + + Γ + + ( + + c + - + + ⅈ + t + + + ) + + + + Γ + + ( + + d + - + + ⅈ + t + + + ) + + + + ⅆ + t + + + + = + + + + Γ + + ( + + a + + + c + + ) + + + + Γ + + ( + + a + + + d + + ) + + + + Γ + + ( + + b + + + c + + ) + + + + Γ + + ( + + b + + + d + + ) + + + + + Γ + + ( + + a + + + b + + + c + + + d + + ) + + + + + + +
+ +
+ + + + + ℜ + a + + , + + ℜ + b + + , + + ℜ + c + + , + + ℜ + d + + + > + 0 + + +
+ +
Ramanujan's Beta Integral
+ +
+ + + + + + ∫ + + - + ∞ + + ∞ + + + + ⅆ + t + + + + Γ + + ( + + a + + + t + + ) + + + + Γ + + ( + + b + + + t + + ) + + + + Γ + + ( + + c + - + t + + ) + + + + Γ + + ( + + d + - + t + + ) + + + + + + = + + + Γ + + ( + + + a + + + b + + + c + + + d + + - + 3 + + ) + + + + + Γ + + ( + + + a + + + c + + - + 1 + + ) + + + + Γ + + ( + + + a + + + d + + - + 1 + + ) + + + + Γ + + ( + + + b + + + c + + - + 1 + + ) + + + + Γ + + ( + + + b + + + d + + - + 1 + + ) + + + + + + + +
+ +
+ + + + ℜ + + ( + + a + + + b + + + c + + + d + + ) + + + > + 3 + + . +
+ +
de Branges-Wilson Beta Integral
+ +
+ + + + + + 1 + + 4 + π + + + ⁢ + + + ∫ + + - + ∞ + + ∞ + + + + + ∏ + + k + = + 1 + + 4 + + + Γ + + ( + + + a + k + + + + + ⅈ + t + + + ) + + + + Γ + + ( + + + a + k + + - + + ⅈ + t + + + ) + + + + + + Γ + + ( + + 2 + ⅈ + t + + ) + + + + Γ + + ( + + - + + 2 + ⅈ + t + + + ) + + + + + + ⅆ + t + + + + = + + + + ∏ + + 1 + ≤ + j + < + k + ≤ + 4 + + + Γ + + ( + + + a + j + + + + + a + k + + + ) + + + + Γ + + ( + + + a + 1 + + + + + a + 2 + + + + + a + 3 + + + + + a + 4 + + + ) + + + + + + +
+ +
+ + + + ℜ + + ( + + a + k + + ) + + + > + 0 + + , + + + k + = + + 1 + , + … + , + 4 + + + . +
+<> +@ + +\subsection{dlmfintegralrepresentations.xhtml} +<>= +<> + + +<> +
+ + Digital Library of Mathematical Functions +
+ The Gamma Function -- Integral Representations +
+
+

Integral Representations

+ +
Contents
+
    +
  • Gamma Function
    • +
  • Psi Function and Euler's Constant
    • +
+ +

Gamma Function

+ +
+ + + + + + 1 + μ + + + Γ + + ( + + ν + μ + + ) + + + + 1 + + z + + ν + μ + + + + + = + + + ∫ + 0 + ∞ + + + exp + + ( + + - + + z + + t + μ + + + + ) + + + + t + + ν + - + 1 + + + + ⅆ + t + + + + + +
+ +

+ + + + ℜ + ν + + > + 0 + + , + + + μ + > + 0 + + , and + + + + ℜ + z + + > + 0 + + . (The fractional powers have their principal values.) +

+ +
Hankel's Loop Integral
+ +
+ + + + + 1 + + Γ + + ( + z + ) + + + + = + + + 1 + + 2 + π + ⅈ + + + + + ∫ + + - + ∞ + + + ( + + 0 + + + + ) + + + + ⅇ + t + + + t + + - + z + + + + ⅆ + t + + + + + + +
+ +

where the contour begins at + + + - + ∞ + + , circles the origin once in the positive direction, and returns to + + + - + ∞ + + . + + + t + + - + z + + + has its principal value where + + t + crosses the positive real axis, and is continuous. +

+ +
+ +
+ +
+ + t + -plane. Contour for Hankel's loop integral. +
+ +
+ + + + + + c + + - + z + + + + Γ + + ( + z + ) + + + + = + + + ∫ + + - + ∞ + + ∞ + + + + | + t + | + + + + 2 + z + + - + 1 + + + + ⅇ + + - + + c + + t + 2 + + + + + + ⅆ + t + + + + + +
+ +
+ + + c + > + 0 + + , + + + + ℜ + z + + > + 0 + + +
+ +

where the path is the real axis. +

+ +
+ + + + + Γ + + ( + z + ) + + + = + + + + ∫ + 1 + ∞ + + + t + + z + - + 1 + + + + ⅇ + + - + t + + + + ⅆ + t + + + + + + + ∑ + + k + = + 0 + + ∞ + + + + + ( + + - + 1 + + ) + + k + + + + ( + + z + + + k + + ) + + + k + ! + + + + + + + + +
+ +
+ + + z + ≠ + + 0 + , + + - + 1 + + , + + - + 2 + + , + … + + + +
+ +
+ + + + + Γ + + ( + z + ) + + + = + + + ∫ + 0 + ∞ + + + t + + z + - + 1 + + + + ( + + + ⅇ + + - + t + + + - + + + ∑ + + k + = + 0 + + n + + + + + + ( + + - + 1 + + ) + + k + + + t + k + + + + k + ! + + + + + ) + + + + ⅆ + t + + + + + +
+ +
+ + + + + - + n + + - + 1 + + < + + ℜ + z + + < + + - + n + + + +
+ +
+ + + + + + Γ + + ( + z + ) + + + + + cos + + ( + + + + 1 + 2 + + + π + z + + ) + + + + = + + + ∫ + 0 + ∞ + + + t + + z + - + 1 + + + + cos + + t + + + + ⅆ + t + + + + + +
+ +
+ + + 0 + < + + ℜ + z + + < + 1 + + , +
+ +
+ + + + + + Γ + + ( + z + ) + + + + + sin + + ( + + + + 1 + 2 + + + π + z + + ) + + + + = + + + ∫ + 0 + ∞ + + + t + + z + - + 1 + + + + sin + + t + + + + ⅆ + t + + + + + +
+ +
+ + + + - + 1 + + < + + ℜ + z + + < + 1 + + . +
+ +
+ + + + + + Γ + + ( + + 1 + + + + 1 + n + + + ) + + + + + cos + + ( + + π + + 2 + n + + + ) + + + + = + + + ∫ + 0 + ∞ + + + cos + + ( + + t + n + + ) + + + + + ⅆ + t + + + + + +
+ +
+ + + n + = + + 2 + , + 3 + , + 4 + , + … + + + +
+ +
+ + + + + + Γ + + ( + + 1 + + + + 1 + n + + + ) + + + + + sin + + ( + + π + + 2 + n + + + ) + + + + = + + + ∫ + 0 + ∞ + + + sin + + ( + + t + n + + ) + + + + + ⅆ + t + + + + + +
+ +
+ + + n + = + + 2 + , + 3 + , + 4 + , + … + + + . +
+ +
Binet's Formula
+ +
+ + + + + ln + + + Γ + + ( + z + ) + + + + = + + + + + ( + + z + - + + + 1 + 2 + + + + ) + + + + ln + + z + + + - + z + + + + + + + 1 + 2 + + + + ln + + ( + + 2 + π + + ) + + + + + + + 2 + + + ∫ + 0 + ∞ + + + + arctan + + ( + + t + z + + ) + + + + + ⅇ + + 2 + π + t + + + - + 1 + + + + + ⅆ + t + + + + + + + +
+ +

where + + + + | + + ph + ⁡ + z + + | + + < + + π + 2 + + + and the inverse tangent has its principal value. +

+ +
+ + + + + ln + + + Γ + + ( + + z + + + 1 + + ) + + + + = + + + - + + γ + z + + + - + + + 1 + + 2 + π + ⅈ + + + + + ∫ + + + - + c + + - + + ∞ + + ⅈ + + + + + - + c + + + + + ∞ + + ⅈ + + + + + + π + + z + + - + s + + + + + s + + + sin + + ( + + π + s + + ) + + + + + + ζ + + ( + + - + s + + ) + + + + + ⅆ + s + + + + + + + +
+ +

where + + + + | + + ph + ⁡ + z + + | + + ≤ + + π + - + δ + + + ( + + + + < + π + + ), + + + 1 + < + c + < + 2 + + , and + + + ζ + + ( + s + ) + + + +

+ +

For additional representations see + + Whittaker and Watson(1927) +

+ +

Psi Function and Euler's Constant

+

For + + + + ℜ + z + + > + 0 + + , +

+ +
+ + + + + ψ + + ( + z + ) + + + = + + + ∫ + 0 + ∞ + + + ( + + + + ⅇ + + - + t + + + t + + - + + + ⅇ + + - + + z + t + + + + + 1 + - + + ⅇ + + - + t + + + + + + ) + + + + ⅆ + t + + + + + +
+ +
+ + + + + ψ + + ( + z + ) + + + = + + + ln + + z + + + + + + ∫ + 0 + ∞ + + + ( + + + 1 + t + + - + + 1 + + 1 + - + + ⅇ + + - + t + + + + + + ) + + + + ⅇ + + - + + t + z + + + + + + ⅆ + t + + + + + + +
+ +
+ + + + + ψ + + ( + z + ) + + + = + + + ∫ + 0 + ∞ + + + ( + + + ⅇ + + - + t + + + - + + 1 + + + ( + + 1 + + + t + + ) + + z + + + + ) + + + + + ⅆ + t + + t + + + + + +
+ +
+ + + + + ψ + + ( + z + ) + + + = + + + ln + + z + + - + + 1 + + 2 + z + + + - + + 2 + + + ∫ + 0 + ∞ + + + + t + + ⅆ + t + + + + + ( + + + t + 2 + + + + + z + 2 + + + ) + + + ( + + + ⅇ + + 2 + π + t + + + - + 1 + + ) + + + + + + + + + +
+ +
+ + + + + + ψ + + ( + z + ) + + + + + γ + + = + + + ∫ + 0 + ∞ + + + + + ⅇ + + - + t + + + - + + ⅇ + + - + + z + t + + + + + + 1 + - + + ⅇ + + - + t + + + + + + + ⅆ + t + + + = + + + ∫ + 0 + 1 + + + + 1 + - + + t + + z + - + 1 + + + + + 1 + - + t + + + + + ⅆ + t + + + + + +
+ +
+ + + + + ψ + + ( + + z + + + 1 + + ) + + + = + + + - + γ + + + + + + 1 + + 2 + π + ⅈ + + + + + ∫ + + + - + c + + - + + ∞ + + ⅈ + + + + + - + c + + + + + ∞ + + ⅈ + + + + + + π + + z + + + - + s + + - + 1 + + + + + sin + + ( + + π + s + + ) + + + + + ζ + + ( + + - + s + + ) + + + + + ⅆ + s + + + + + + + +
+ +

where + + + + | + + ph + + z + + | + + ≤ + + + π + - + δ + + + ( + + + < + π + + ) + + + + and + + + 1 + < + c + < + 2 + + . +

+ +
+ + + + γ + = + + - + + + ∫ + 0 + ∞ + + + ⅇ + + - + t + + + + + ln + + t + + + + ⅆ + t + + + + = + + + ∫ + 0 + ∞ + + + ( + + + 1 + + 1 + + + t + + + - + + ⅇ + + - + t + + + + ) + + + + + ⅆ + t + + t + + + = + + + + ∫ + 0 + 1 + + + ( + + 1 + - + + ⅇ + + - + t + + + + ) + + + + + ⅆ + t + + t + + + - + + + ∫ + 1 + ∞ + + + ⅇ + + - + t + + + + + ⅆ + t + + t + + + + = + + + ∫ + 0 + ∞ + + + ( + + + + ⅇ + + - + t + + + + 1 + - + + ⅇ + + - + t + + + + + - + + + ⅇ + + - + t + + + t + + + ) + + + + ⅆ + t + + + + + +
+<> +@ + +\subsection{dlmfmathematicalapplications.xhtml} +<>= +<> + + +<> +
+ + Digital Library of Mathematical Functions +
+ The Gamma Function -- Mathematical Applications +
+
+

Mathematical Applications

+
Contents
+
    +
  • Summation of Rational Functions
    • +
  • Mellin-Barnes Integrals
    • +
  • + + n + -Dimensional Sphere
    • +
+ +

Summation of Rational Functions

+ +

As shown in + + Temme(1996) + (§3.4), the results given in + + Series Expansions + can be used to sum infinite series of rational functions. +

+ +
Example
+ +
+ + + + S + = + + + + ∑ + + k + = + 0 + + ∞ + + + + a + k + + + + , + + + + + + + a + k + + = + + + k + + + ( + + + 3 + k + + + + 2 + + ) + + + ( + + + 2 + k + + + + 1 + + ) + + + ( + + k + + + 1 + + ) + + + + + + + +
+ +

By decomposition into partial fractions

+ +
+ + + + + a + k + + = + + + 2 + + k + + + + 2 + 3 + + + + - + + 1 + + k + + + + 1 + 2 + + + + - + + 1 + + k + + + 1 + + + + = + + + ( + + + 1 + + k + + + 1 + + + - + + 1 + + k + + + + 1 + 2 + + + + + ) + + - + + 2 + + ( + + + 1 + + k + + + 1 + + + - + + 1 + + k + + + + 2 + 3 + + + + + ) + + + + + + +
+ +

Hence from ( + Series Expansions 6 + ), ( Special Values and Extrema + + Equation 13 + and + + Equation 19 + ) +

+ +
+ + + + S + = + + + ψ + + ( + + + 1 + 2 + + + ) + + + - + + 2 + + ψ + + ( + + + 2 + 3 + + + ) + + + + - + γ + + = + + + 3 + + ln + 3 + + + - + + 2 + + ln + 2 + + + - + + + + 1 + 3 + + + π + + 3 + + + + + + +
+ +

Mellin-Barnes Integrals

+

Many special functions + + + f + + ( + z + ) + + + can be represented as a Mellin-Barnes integral, that is, + an integral of a product of gamma functions, reciprocals of gamma + functions, and a power of + + z + , the integration contour being doubly-infinite and eventually + parallel to the imaginary axis. The left-hand side of ( + + Integral Equation 1 + ) is a typical example. By translating the contour parallel to itself + and summing the residues of the integrand, asymptotic expansions of + + + f + + ( + z + ) + + + for large + + + | + z + | + + , or small + + + | + z + | + + , can be obtained complete with an integral representation of the + error term. +

+ +

+ + n + -Dimensional Sphere

+ +

The volume + + V + and surface area + + A + of the + + n + -dimensional sphere of radius + + r + are given by +

+ +
+ + + + V + = + + + + + π + + + 1 + 2 + + n + + + + r + n + + + + Γ + + ( + + + + 1 + 2 + + n + + + + 1 + + ) + + + + + + , + + + + + + S + = + + + + 2 + + π + + + 1 + 2 + + n + + + + r + + n + - + 1 + + + + + Γ + + ( + + + 1 + 2 + + n + + ) + + + + + = + + + + n + r + + + V + + + + +
+<> +@ + +\subsection{dlmfmethodsofcomputation.xhtml} +<>= +<> + + +<> +
+ + Digital Library of Mathematical Functions +
+ The Gamma Function -- Methods of Computation +
+
+

Methods of Computation

+ +

An effective way of computing + + + Γ + + ( + z + ) + + + +in the right half-plane is backward recurrence, beginning with a value +generated from the + + asymptotic expansion + +Or we can use forward recurrence, with an + + initial value +. +For the left half-plane we can continue the backward recurrence or +make use of the + + reflection formula +. +

+ +

Similarly for + + + ln + + Γ + + ( + z + ) + + + + , + + + ψ + + ( + z + ) + + + , and the polygamma functions. +

+ +

For a comprehensive survey see + + van der Laan and Temme(1984) + (Chapter III). + See also + + Borwein and Zucker(1992) + . +

+<> +@ + +\subsection{dlmfmultidimensionalintegral.xhtml} +<>= +<> + + +<> +
+ + Digital Library of Mathematical Functions +
+ The Gamma Function -- Multidimensional Integral +
+
+

Multidimensional Integrals

+ +

Let + + + V + n + + be the simplex: + + + + + t + 1 + + + + + t + 2 + + + + … + + + + t + n + + + ≤ + 1 + + , + + + + t + k + + ≥ + 0 + + . Then for + + + + ℜ + + z + k + + + > + 0 + + , + + + k + = + + 1 + , + 2 + , + … + , + + n + + + 1 + + + + , +

+ +
+ + + + + + ∫ + + V + n + + + + t + 1 + + + z + 1 + + - + 1 + + + + t + 2 + + + z + 2 + + - + 1 + + + ⋯ + + t + n + + + z + n + + - + 1 + + + + ⅆ + + t + 1 + + + + ⅆ + + t + 2 + + + ⋯ + + ⅆ + + t + n + + + + = + + + + Γ + + ( + + z + 1 + + ) + + + + Γ + + ( + + z + 2 + + ) + + + ⋯ + + Γ + + ( + + z + n + + ) + + + + + Γ + + ( + + 1 + + + + z + 1 + + + + + z + 2 + + + + … + + + + z + n + + + ) + + + + + + +
+ +
+ + + + + + ∫ + + V + n + + + + + ( + + 1 + - + + + ∑ + + k + = + 1 + + n + + + t + k + + + + ) + + + + z + + n + + + 1 + + + - + 1 + + + + + ∏ + + k + = + 1 + + n + + + t + k + + + z + k + + - + 1 + + + + ⅆ + + t + k + + + + + = + + + + Γ + + ( + + z + 1 + + ) + + + + Γ + + ( + + z + 2 + + ) + + + ⋯ + + Γ + + ( + + z + + n + + + 1 + + + ) + + + + + Γ + + ( + + + z + 1 + + + + + z + 2 + + + + … + + + + z + + n + + + 1 + + + + ) + + + + + + +
+ +
Selberg-type Integrals
+ +
+ + + + + Δ + + ( + + + t + 1 + + , + + t + 2 + + , + … + , + + t + n + + + ) + + + = + + + ∏ + + 1 + ≤ + j + < + k + ≤ + n + + + ( + + + t + j + + - + + t + k + + + ) + + + + +
+ +

Then +

+ +
+ + + + + + ∫ + + + [ + + 0 + , + 1 + + ] + + n + + + + t + 1 + + + t + 2 + + ⋯ + + t + m + + + + | + + Δ + + ( + + + t + 1 + + , + … + , + + t + n + + + ) + + + | + + + 2 + c + + + + + ∏ + + k + = + 1 + + n + + + t + k + + a + - + 1 + + + + + ( + + 1 + - + + t + k + + + ) + + + b + - + 1 + + + + ⅆ + + t + k + + + + + = + + + 1 + + + ( + + Γ + + ( + + 1 + + + c + + ) + + + ) + + n + + + + + ∏ + + k + = + 1 + + m + + + + a + + + + + ( + + n + - + k + + ) + + c + + + + a + + + b + + + + + ( + + + 2 + n + + - + k + - + 1 + + ) + + c + + + + + + ∏ + + k + = + 1 + + n + + + + + Γ + + ( + + a + + + + + ( + + n + - + k + + ) + + c + + + ) + + + + Γ + + ( + + b + + + + + ( + + n + - + k + + ) + + c + + + ) + + + + Γ + + ( + + 1 + + + + k + c + + + ) + + + + + Γ + + ( + + a + + + b + + + + + ( + + + 2 + n + + - + k + - + 1 + + ) + + c + + + ) + + + + + + + + + +
+ +

provided that + + + ℜ + a + + , + + + + ℜ + b + + > + 0 + + , + + + + ℜ + c + + > + + - + + min + + ( + + + 1 + n + + , + + ℜ + + a + + ( + + n + - + 1 + + ) + + + + , + + ℜ + + b + + ( + + n + - + 1 + + ) + + + + + ) + + + + + +

+ +

Secondly, +

+ +
+ + + + + + ∫ + + + [ + + 0 + , + ∞ + + ) + + n + + + + t + 1 + + + t + 2 + + ⋯ + + t + m + + + + | + + Δ + + ( + + + t + 1 + + , + … + , + + t + n + + + ) + + + | + + + 2 + c + + + + + ∏ + + k + = + 1 + + n + + + t + k + + a + - + 1 + + + + ⅇ + + - + + t + k + + + + + ⅆ + + t + k + + + + + = + + + ∏ + + k + = + 1 + + m + + + ( + + a + + + + + ( + + n + - + k + + ) + + c + + + ) + + + + + ∏ + + k + = + 1 + + n + + + Γ + + + ( + + a + + + + + ( + + n + - + k + + ) + + c + + + ) + + + + Γ + + ( + + 1 + + + + k + c + + + ) + + + + + + ( + + Γ + + ( + + 1 + + + c + + ) + + + ) + + n + + + + + + +
+ +

when + + + + ℜ + a + + > + 0 + + , + + + + ℜ + c + + > + + - + + min + + ( + + + 1 + n + + , + + ℜ + + a + + ( + + n + - + 1 + + ) + + + + + ) + + + + + . +

+ +

Thirdly, +

+ +
+ + + + + + 1 + + + ( + + 2 + π + + ) + + + n + 2 + + + + + + ∫ + + + ( + + + - + ∞ + + , + ∞ + + ) + + n + + + + + | + + Δ + + ( + + + t + 1 + + , + … + , + + t + n + + + ) + + + | + + + 2 + c + + + + + ∏ + + k + = + 1 + + n + + + exp + + ( + + - + + + + 1 + 2 + + + + t + k + 2 + + + + ) + + + + ⅆ + + t + k + + + + + + = + + + + ∏ + + k + = + 1 + + n + + Γ + + ( + + 1 + + + + k + c + + + ) + + + + + ( + + Γ + + ( + + 1 + + + c + + ) + + + ) + + n + + + + + +
+ +
Dyson's Integral
+ +
+ + + + + + 1 + + + ( + + 2 + π + + ) + + n + + + + + ∫ + + + [ + + + - + π + + , + π + + ] + + n + + + + ∏ + + 1 + ≤ + j + < + k + ≤ + n + + + + + | + + + ⅇ + + ⅈ + + θ + j + + + + - + + ⅇ + + ⅈ + + θ + k + + + + + | + + + 2 + b + + + + ⅆ + + θ + 1 + + + ⋯ + + ⅆ + + θ + n + + + + + = + + + Γ + + ( + + 1 + + + + b + n + + + ) + + + + + ( + + Γ + + ( + + 1 + + + b + + ) + + + ) + + n + + + + + +
+ +
+ + + + ℜ + b + + > + + + 1 + n + + + + . +
+<> +@ + +\subsection{dlmfnotation.xhtml} +<>= +<> + + +<> +
Digital Library of Mathematical Functions
+ The Gamma Function -- Notation +
+
+
+
+

Notation

+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + + j + , + m + , + n + + + nonnegative integers.
+ + k + + except in + Physical Applications +
+ + + x + , + y + + + real variables.
+ + + z + = + + x + + + + ⅈ + y + + + + + complex variable.
+ + + a + , + b + , + q + , + s + , + w + + + real or complex variables with + + + + ∣ + q + ∣ + + < + 1 + + . +
+ + δ + + arbitrary small positive constant.
+ + ℂ + + complex plane (excluding infinity).
+ + ℝ + + real line (excluding infinity).
+ + + ( + + + + + n + + + + + m + + + + + ) + + + binomial coefficient + + + + n + ! + + + + m + ! + + + + ( + + n + - + m + + ) + + ! + + + + . +
empty sumszero.
empty productsunity.
+
+ +
+

The main functions treated in this chapter are the gamma function + + + Γ + + ( + z + ) + + + ,the psi function + + + ψ + + ( + z + ) + + + ,the beta function + + + B + + ( + + a + , + b + + ) + + + , and the + + q + -gamma function + + + + Γ + q + + + ( + z + ) + + + . +

+
+ +
+

The notation + + + Γ + + ( + z + ) + + + is due to Legendre. Alternative notations for this function are: + + + Π + + ( + + z + - + 1 + + ) + + + (Gauss) and + + + + ( + + z + - + 1 + + ) + + ! + + . Alternative notations for the psi function are: +

+
+ +
+ + + + + + + + + + + + + + + + + + + + + +
+ + + Ψ + + ( + + z + - + 1 + + ) + + + + Gauss; + + Jahnke and Emde(1945) + +
+ + + Ψ + + ( + z + ) + + + + + + Whittaker and Watson(1927) + +
+ + + Ψ + + ( + z + ) + + + + + + Davis(1933) + +
+ + + F + + ( + + z + - + 1 + + ) + + + + + + Pairman(1919) + +
+
+
+
+<> +@ + +\subsection{dlmfphysicalapplications.xhtml} +<>= +<> + + +<> +
+ + Digital Library of Mathematical Functions +
+ The Gamma Function -- Physical Applications +
+
+

Physical Applications

+ +

Suppose the potential energy of a gas of + + n + point charges with positions + + + + x + 1 + + , + + x + 2 + + , + … + , + + x + n + + + and free to move on the infinite line + + + + - + ∞ + + < + x + < + ∞ + + , is given by +

+ +
+ + + + W + = + + + + 1 + 2 + + + + ∑ + + ℓ + = + 1 + + n + + + x + ℓ + 2 + + + + - + + + ∑ + + 1 + ≤ + ℓ + < + j + ≤ + n + + + ln + + | + + + x + ℓ + + - + + x + j + + + | + + + + + + +
+ +

The probability density of the positions when the gas is in thermodynamic + equilibrium is: +

+ +
+ + + + + P + + ( + + + x + 1 + + , + … + , + + x + n + + + ) + + + = + + C + + exp + + ( + + - + + W + + ( + + k + T + + ) + + + + ) + + + + + + +
+ +

where + + k + is the Boltzmann constant, + + T + the temperature and + + C + a constant. + Then the partition function (with + + + β + = + + 1 + + ( + + k + T + + ) + + + + ) is given by +

+ +
+ + + + + + ψ + n + + + ( + β + ) + + + = + + + ∫ + + ℝ + n + + + + ⅇ + + - + + β + W + + + + + ⅆ + x + + + = + + + + + + ( + + 2 + π + + ) + + + n + 2 + + + + β + + + - + + ( + + n + 2 + + ) + + + - + + ( + + + β + n + + ( + + n + - + 1 + + ) + + + 4 + + ) + + + + + × + + + ( + + Γ + + ( + + 1 + + + + + + 1 + 2 + + + β + + + ) + + + ) + + + - + n + + + + + + ∏ + + j + = + 1 + + n + + Γ + + ( + + 1 + + + + + + 1 + 2 + + + j + β + + + ) + + + + + + +
+ +

For + + n + charges free to move on a circular wire of radius + + 1 + , +

+ +
+ + + + W + = + + - + + + ∑ + + 1 + ≤ + ℓ + < + j + ≤ + n + + + ln + + | + + + ⅇ + + ⅈ + + θ + ℓ + + + + - + + ⅇ + + ⅈ + + θ + j + + + + + | + + + + + + +
+ +

and the partition function is given by

+ +
+ + + + + + ψ + n + + + ( + β + ) + + + = + + + 1 + + + ( + + 2 + π + + ) + + n + + + + + ∫ + + + [ + + + - + π + + , + π + + ] + + n + + + + ⅇ + + - + + β + W + + + + + ⅆ + + θ + 1 + + + ⋯ + + ⅆ + + θ + n + + + + + = + + + Γ + + ( + + 1 + + + + + + 1 + 2 + + + n + β + + + ) + + + + + + Γ + + ( + + 1 + + + + + + 1 + 2 + + + β + + + ) + + + ) + + + - + n + + + + + + +
+<> +@ + +\subsection{dlmfpolygammafunctions.xhtml} +<>= +<> + + +<> +
+ + Digital Library of Mathematical Functions +
+ The Gamma Function -- Polygamma Functions +
+
+

Polygamma Functions

+ +

The functions + + + + ψ + + ( + n + ) + + + + ( + z + ) + + + , + + + n + = + + 1 + , + 2 + , + … + + + , are called the polygamma functions. In particular, + + + + ψ + ′ + + + ( + z + ) + + + is the trigamma function; + + + ψ + + ′ + ′ + + , + + + ψ + + ( + 3 + ) + + + , + + + ψ + + ( + 4 + ) + + + are the tetra-, penta-, and + hexagamma functions respectively. Most properties of these + functions follow straightforwardly by differentiation of properties + of the psi function. This includes asymptotic expansions. +

+ +

In the second and third equations, + + + n + = + + 1 + , + 2 + , + 3 + , + … + + + ; for + + + ζ + + ( + + n + + + 1 + + ) + + + +

+ +
+ + + + + + ψ + ′ + + + ( + z + ) + + + = + + + ∑ + + k + = + 0 + + ∞ + + + 1 + + + ( + + k + + + z + + ) + + 2 + + + + + + +
+ +
+ + + z + ≠ + + 0 + , + + - + 1 + + , + + - + 2 + + , + … + + + +
+ +
+ + + + + + ψ + + ( + n + ) + + + + ( + 1 + ) + + + = + + + + ( + + - + 1 + + ) + + + n + + + 1 + + + + n + ! + + + ζ + + ( + + n + + + 1 + + ) + + + + + + +
+ +
+ + + + + + ψ + + ( + n + ) + + + + ( + + + 1 + 2 + + + ) + + + = + + + + ( + + - + 1 + + ) + + + n + + + 1 + + + + n + ! + + + ( + + + 2 + + n + + + 1 + + + - + 1 + + ) + + + ζ + + ( + + n + + + 1 + + ) + + + + + + +
+ +
+ + + + + + ψ + ′ + + + ( + + n + + + + + 1 + 2 + + + + ) + + + = + + + + + 1 + 2 + + + + π + 2 + + + - + + 4 + + + ∑ + + k + = + 1 + + n + + + 1 + + + ( + + + 2 + k + + - + 1 + + ) + + 2 + + + + + + + + +
+ +

As + + + z + → + ∞ + + in + + + + | + + ph + + z + + | + + ≤ + + + π + - + δ + + + ( + + + < + π + + ) + + + + +

+ +
+ + + + + + ψ + ′ + + + ( + z + ) + + + ∼ + + + 1 + z + + + + + 1 + + 2 + + z + 2 + + + + + + + + ∑ + + k + = + 1 + + ∞ + + + + B + + 2 + k + + + + z + + + 2 + k + + + + 1 + + + + + + + + +
+<> +@ + +\subsection{dlmfqgammaandbetafunctions.xhtml} +<>= +<> + + +<> +
+ + Digital Library of Mathematical Functions +
+ The Gamma Function -- q-Gamma and Beta Functions +
+
+

+ + q + -Gamma and Beta Functions +

+ +
    +
  • + + q + -Factorials
    • +
  • + + q + -Gamma Function
    • +
  • + + q + -Beta Function
    • +
+ +

+ + q + -Factorials

+ +
+ + + + + + ( + + a + + ; + q + + ) + + n + + = + + + ∏ + + k + = + 0 + + + n + - + 1 + + + ( + + 1 + - + + a + + q + k + + + + ) + + + + +
+ +
+ + + n + = + + 0 + , + 1 + , + 2 + , + … + + + +
+ +
+ + + + + + n + ! + + q + + = + + 1 + + ( + + 1 + + + q + + ) + + ⋯ + + ( + + 1 + + + q + + + … + + + + q + + n + - + 1 + + + + ) + + + = + + + + ( + + q + + ; + q + + ) + + n + + + + ( + + 1 + - + q + + ) + + + - + n + + + + + + +
+ +

When + + + + | + q + | + + < + 1 + + , +

+ +
+ + + + + + ( + + a + + ; + q + + ) + + ∞ + + = + + + ∏ + + k + = + 0 + + ∞ + + ( + + 1 + - + + a + + q + k + + + + ) + + + + +
+ +

+ + q-Gamma Function

+ +

When + + + 0 + < + q + < + 1 + + , +

+ +
+ + + + + + Γ + q + + + ( + z + ) + + + = + + + + + ( + + q + + ; + q + + ) + + ∞ + + + + ( + + 1 + - + q + + ) + + + 1 + - + z + + + + + + ( + + + q + z + + + ; + q + + ) + + ∞ + + + + + +
+ +
+ + + + + + Γ + q + + + ( + 1 + ) + + + = + + + Γ + q + + + ( + 2 + ) + + + = + 1 + + + +
+ +
+ + + + + + n + ! + + q + + = + + + Γ + q + + + ( + + n + + + 1 + + ) + + + + + +
+ +
+ + + + + + Γ + q + + + ( + + z + + + 1 + + ) + + + = + + + + 1 + - + + q + z + + + + 1 + - + q + + + + + Γ + q + + + ( + z + ) + + + + + + +
+ +

Also, + + + ln + + + Γ + q + + + ( + x + ) + + + + is convex for + + + x + > + 0 + + , and the analog of the + + Bohr-Mollerup theorem + holds. +

+ +

If + + + 0 + < + q + < + r + < + 1 + + , then +

+ +
+ + + + + + Γ + q + + + ( + x + ) + + + < + + + Γ + r + + + ( + x + ) + + + + + +
+ +

when + + + 0 + < + x + < + 1 + + or when + + + x + > + 2 + + , and +

+ +
+ + + + + + Γ + q + + + ( + x + ) + + + > + + + Γ + r + + + ( + x + ) + + + + + +
+ +

when + + + 1 + < + x + < + 2 + + . +

+ +
+ + + + + + lim + + q + → + + 1 + - + + + + + + Γ + q + + + ( + z + ) + + + + = + + Γ + + ( + z + ) + + + + + +
+ +

For generalized asymptotic expansions of + + + ln + + + + Γ + q + + + ( + z + ) + + + + as + + + + | + z + | + + → + ∞ + + see + + Olde Daalhuis(1994) + and + + Moak(1984) + . +

+ +

+ + q + -Beta Function +

+ +
+ + + + + + B + q + + + ( + + a + , + b + + ) + + + = + + + + + Γ + q + + + ( + a + ) + + + + + Γ + q + + + ( + b + ) + + + + + + Γ + q + + + ( + + a + + + b + + ) + + + + + + +
+ +
+ + + + + + B + q + + + ( + + a + , + b + + ) + + + = + + + ∫ + 0 + 1 + + + + + t + + a + - + 1 + + + + + ( + + + t + q + + + ; + q + + ) + + ∞ + + + + + ( + + + t + + q + b + + + ; + q + + ) + + ∞ + + + + + ⅆ + q + + t + + + + + +
+ +
+ + + 0 + < + q + < + 1 + + , + + + + ℜ + a + + > + 0 + + , + + + + ℜ + b + + > + 0 + + . +
+<> +@ + +\subsection{dlmfseriesexpansions.xhtml} +<>= +<> + + +<> +
+ + Digital Library of Mathematical Functions +
+ The Gamma Function -- Series Expansions +
+
+

Series Expansions

+
Contents
+
    +
  • Maclaurin Series
    • +
  • Other Series
    • +
+

Maclaurin Series

+

Throughout this subsection + + + ζ + + ( + k + ) + + + is +

+ +
+ + + + + 1 + + Γ + + ( + z + ) + + + + = + + + ∑ + + k + = + 1 + + ∞ + + + c + k + + + z + k + + + + + +
+ +

where + + + + c + 1 + + = + 1 + + , + + + + + c + 2 + + = + γ + + , and +

+ +
+ + + + + + ( + + k + - + 1 + + ) + + + c + k + + + = + + + + + + γ + + c + + k + - + 1 + + + + - + + + ζ + + ( + 2 + ) + + + + c + + k + - + 2 + + + + + + + + + ζ + + ( + 3 + ) + + + + c + + k + - + 3 + + + + + - + … + + + + + + + ( + + - + 1 + + ) + + k + + + ζ + + ( + + k + - + 1 + + ) + + + + c + 1 + + + + + + +
+ +
+ + + k + ≥ + 3 + + . +
+ +

For 15D numerical values of + + + c + k + + see + + Abramowitz and Stegun(1964)(p. 256), and +for 31D values see + + Wrench(1968). +

+ + +
+ + + + + ln + + Γ + + ( + + 1 + + + z + + ) + + + + = + + + - + + ln + + ( + + 1 + + + z + + ) + + + + + + + z + + ( + + 1 + - + γ + + ) + + + + + + + ∑ + + k + = + 2 + + ∞ + + + + ( + + - + 1 + + ) + + k + + + ( + + + ζ + + ( + k + ) + + + - + 1 + + ) + + + + z + k + + k + + + + + + +
+ +
+ + + + | + z + | + + < + 2 + + . +
+ +
+ + + + + ψ + + ( + + 1 + + + z + + ) + + + = + + + - + γ + + + + + + ∑ + + k + = + 2 + + ∞ + + + + ( + + - + 1 + + ) + + k + + + ζ + + ( + k + ) + + + + z + + k + - + 1 + + + + + + + +
+ +
+ + + + | + z + | + + < + 1 + + , +
+ +
+ + + + + ψ + + ( + + 1 + + + z + + ) + + + = + + + + + 1 + + 2 + z + + + - + + + π + 2 + + + cot + + ( + + π + z + + ) + + + + + + + + 1 + + + z + 2 + + - + 1 + + + + + 1 + + - + γ + - + + + ∑ + + k + = + 1 + + ∞ + + + ( + + + ζ + + ( + + + 2 + k + + + + 1 + + ) + + + - + 1 + + ) + + + z + + 2 + k + + + + + + + +
+ +
+ + + + | + z + | + + < + 2 + + , + + + z + ≠ + + 0 + , + + ± + 1 + + + + . +
+ +

For 20D numerical values of the coefficients of the Maclaurin series for + + + Γ + + ( + + z + + + 3 + + ) + + + see + + Luke(1969)(p. 299). +

+ +

When + + + z + ≠ + + 0 + , + + - + 1 + + , + + - + 2 + + , + … + + + , +

+ + +
+ + + + + ψ + + ( + z + ) + + + = + + + + - + γ + + - + + 1 + z + + + + + + + ∑ + + k + = + 1 + + ∞ + + + z + + k + + ( + + k + + + z + + ) + + + + + + = + + + - + γ + + + + + + ∑ + + k + = + 0 + + ∞ + + ( + + + 1 + + k + + + 1 + + + - + + 1 + + k + + + z + + + + ) + + + + + +
+ +

and +

+ +
+ + + + + + ψ + + ( + + + z + + + 1 + + 2 + + ) + + + - + + ψ + + ( + + z + 2 + + ) + + + + = + + 2 + + + ∑ + + k + = + 0 + + ∞ + + + + + ( + + - + 1 + + ) + + k + + + k + + + z + + + + + + + +
+ +

Also, +

+ +
+ + + + + ℑ + + ψ + + ( + + + ⅈ + y + + + + 1 + + ) + + + + = + + + ∑ + + k + = + 1 + + ∞ + + + y + + + k + 2 + + + + + y + 2 + + + + + + + +
+<> +@ + +\subsection{dlmfsums.xhtml} +<>= +<> + + +<> + +
+

Sums

+ +
+ + + + + + ∑ + + k + = + 1 + + ∞ + + + + ( + + - + 1 + + ) + + k + + + + ψ + ′ + + + ( + k + ) + + + + = + + - + + + π + 2 + + 8 + + + + + +
+ +
+ + + + + + ∑ + + k + = + 1 + + ∞ + + + 1 + k + + + + ψ + ′ + + + ( + + k + + + 1 + + ) + + + + = + + ζ + + ( + 3 + ) + + + = + + - + + + 1 + 2 + + + + ψ + + ′ + ′ + + + + ( + 1 + ) + + + + + + + +
+ +

For further sums involving the psi function see + + Hansen(1975) +(pp. 360–367). For sums of gamma functions see + + Andrews et.al.(1999) +(Chapters 2 and 3). +

+ +

For related sums involving finite field analogs of the gamma and +beta functions (Gauss and Jacobi sums) see + + Andrews et.al.(1999) +(Chapter 1) and + + Terras(1999) +. +

+<> +@ + +\subsection{dlmfsoftware.xhtml} +<>= +<> + + +<> +
+ + Digital Library of Mathematical Functions +
+ The Gamma Function -- Software +
+
+<> +@ + +\subsection{dlmfspecialvaluesandextrema.xhtml} +<>= +<> + + +<> +
+ + Digital Library of Mathematical Functions +
+ The Gamma Function -- Special Values and Extrema +
+
+

Special Values and Extrema

+
Contents
+
    +
  • Gamma Function
    • +
  • Psi Function
    • +
  • Extrema
    • +
+ +

Gamma Function

+
+ + + + + Γ + + ( + 1 + ) + + + = + 1 + + + +
+ +
+ + + + + Γ + + ( + + n + + + 1 + + ) + + + = + + n + ! + + + + +
+ +
+ + + + + ∣ + + Γ + + ( + + ⅈ + y + + ) + + + ∣ + + = + + + ( + + π + + y + + sinh + + ( + + π + y + + ) + + + + + ) + + + 1 + 2 + + + + + +
+ +
+ + + + + + Γ + + ( + + + + 1 + 2 + + + + + + ⅈ + y + + + ) + + + + Γ + + ( + + + + 1 + 2 + + + - + + ⅈ + y + + + ) + + + + = + + + ∣ + + Γ + + ( + + + + 1 + 2 + + + + + + ⅈ + y + + + ) + + + ∣ + + 2 + + = + + π + + cosh + + ( + + π + y + + ) + + + + + + +
+ +
+ + + + + + Γ + + ( + + + + 1 + 4 + + + + + + ⅈ + y + + + ) + + + + Γ + + ( + + + + 3 + 4 + + + - + + ⅈ + y + + + ) + + + + = + + + π + + 2 + + + + + cosh + + ( + + π + y + + ) + + + + + + ⅈ + + sinh + + ( + + π + y + + ) + + + + + + + + +
+ +
+ + + + + Γ + + ( + + + 1 + 2 + + + ) + + + = + + π + + 1 + 2 + + + = + + 1.77245 38509 05516 02729 + … + + + + +
+ +
+ + + + + Γ + + ( + + + 1 + 3 + + + ) + + + = + + 2.67893 85347 07747 63365 + … + + + + +
+ +
+ + + + + Γ + + ( + + + 2 + 3 + + + ) + + + = + + 1.35411 79394 26400 41694 + … + + + + +
+ +
+ + + + + Γ + + ( + + + 1 + 4 + + + ) + + + = + + 3.62560 99082 21908 31193 + … + + + + +
+ +
+ + + + + Γ + + ( + + + 3 + 4 + + + ) + + + = + + 1.22541 67024 65177 64512 + … + + + + +
+ +
+ + + + + + Γ + ′ + + + ( + 1 + ) + + + = + + - + γ + + + + +
+ +

Psi Function

+
+ + + + + ψ + + ( + 1 + ) + + + = + + - + γ + + + + +
+ + +
+ + + + + ψ + + ( + + + 1 + 2 + + + ) + + + = + + + - + γ + + - + + 2 + + ln + 2 + + + + + + +
+ +
+ + + + + ψ + + ( + + n + + + 1 + + ) + + + = + + + + ∑ + + k + = + 1 + + n + + + 1 + k + + + - + γ + + + + +
+ +
+ + + + + ψ + + ( + + n + + + + + 1 + 2 + + + + ) + + + = + + + + - + γ + + - + + 2 + + ln + 2 + + + + + + + 2 + + ( + + 1 + + + + + 1 + 3 + + + + + … + + + + + 1 + + + 2 + n + + - + 1 + + + + + ) + + + + + + + + + n + ≥ + 1 + + +
+ +
+ + + + + ℑ + + ψ + + ( + + ⅈ + y + + ) + + + + = + + + 1 + + 2 + y + + + + + + + π + 2 + + + coth + + ( + + π + y + + ) + + + + + + + +
+ +
+ + + + + ℑ + + ψ + + ( + + + + 1 + 2 + + + + + + ⅈ + y + + + ) + + + + = + + + π + 2 + + + tanh + + ( + + π + y + + ) + + + + + + +
+ +
+ + + + + ℑ + + ψ + + ( + + 1 + + + + ⅈ + y + + + ) + + + + = + + + - + + 1 + + 2 + y + + + + + + + + π + 2 + + + coth + + ( + + π + y + + ) + + + + + + + +
+ +

+ + + 0 + < + p + < + q + + are integers, then +

+ + +
+ + + + + ψ + + ( + + p + q + + ) + + + = + + + + - + γ + + - + + ln + q + + - + + + π + 2 + + + cot + + ( + + + π + p + + q + + ) + + + + + + + + + 1 + 2 + + + + ∑ + + k + = + 1 + + + q + - + 1 + + + + cos + + ( + + + 2 + π + k + p + + q + + ) + + + + ln + + ( + + 2 + - + + 2 + + cos + + ( + + + 2 + π + k + + q + + ) + + + + + ) + + + + + + + + +
+ +

Extrema

+
+ + + + + Γ + ′ + + + ( + + x + n + + ) + + + = + + ψ + + ( + + x + n + + ) + + + = + 0 + + . +
+
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + n + + + + + x + n + + + + + + Γ + + ( + + x + n + + ) + + + +
0 + + + 1.46163 21449 + + + + 0.88560 31944 + +
1 + + + + - + 0.50408 30083 + + + + + + - + 3.54464 36112 + + +
2 + + + + - + 1.57349 84732 + + + + + 2.30240 72583 + +
3 + + + + - + 2.61072 08875 + + + + + + - + 0.88813 63584 + + +
4 + + + + - + 3.63529 33665 + + + + + 0.24512 75398 + +
5 + + + + - + 4.65323 77626 + + + + + + - + 0.05277 96396 + + +
6 + + + + - + 5.66716 24513 + + + + + 0.00932 45945 + +
7 + + + + - + 6.67841 82649 + + + + + + - + 0.00139 73966 + + +
8 + + + + - + 7.68778 83250 + + + + + 0.00018 18784 + +
9 + + + + - + 8.69576 41633 + + + + + + - + 0.00002 09253 + + +
10 + + + + - + 9.70267 25406 + + + + + 0.00000 21574 + +
+
+ +

As + + + n + ∞ + + , +

+ +
+ + + + + x + n + + = + + + - + n + + + + + + 1 + π + + + arctan + + ( + + π + + ln + n + + + ) + + + + + + + O + + ( + + 1 + + n + + + ( + + ln + n + + ) + + 2 + + + + ) + + + + + + +
+<> +@ + +\subsection{dlmftables.xhtml} +<>= +<> + + +<> + +
+

Tables

+ +These tables show Axiom's compliance with published standard values. +In all cases shown here Axiom conforms to the accuracy of the published +tables. + +
    +
  • The Gamma Function
    • +
  • The Psi Function
    • +
+ +

The Gamma Function

+ +This table was constructed from the published values in the +Handbook of Mathematical Functions, by Milton Abramowitz +and Irene A. Stegun, by Dover (1965), pp 267-270. + +The first column is the point where the Gamma function is evaluated. +The second column is the value reported in the Handbook. +The third column is the actual value computed by Axiom at the given point. +The fourth column is the difference of Axiom's value and the Handbook value. + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
pointHandbook ValueAxiom Computed ValueDifference
1.0001.00000000001.0.
1.0050.99713853540.9971385352483757-1.51E-10
1.0100.99432585120.99432585118631189-2.03E-11
1.0150.99156128880.991561288841313234.14E-11
1.0200.98884420330.9888442032538789-4.31E-11
1.0250.98617396330.98617396313592742-1.54E-10
1.0300.98354995060.98354995053928918-7.59E-11
1.0350.98097156060.98097156056367696-4.60E-11
1.0400.97843820090.9784382009247683 3.00E-11
1.0450.97594929190.97594929183099266-6.55E-11
1.0500.97350426560.97350426556841785-2.72E-11
1.0550.97110256630.97110256624499502-6.77E-11
1.0600.96874364950.96874364951272707-2.36E-12
1.0650.96642698230.96642698229777113-1.37E-11
1.0700.96415204250.96415204253821729 4.61E-11
1.0750.96191831890.96191831892929192 2.31E-11
1.0800.95972531070.95972531067573963-3.00E-11
1.0850.95757252730.95757252725116249-3.68E-11
1.0900.95545948820.95545948816407866-4.24E-11
1.0950.95338572270.95338572273049704 2.34E-11
1.1000.95135076990.95135076987625944-2.49E-11
1.1050.94935417780.94935417782771081 2.11E-11
1.1100.94739550400.94739550404472173 5.80E-11
1.1150.94547431490.94547431492209555 1.12E-11
1.1200.94359018560.94359018561564112 1.06E-11
1.1250.94174269970.94174269984970138 1.39E-10
1.1300.93993144970.93993144972988807 1.67E-11
1.1350.93815603560.93815603556085947-5.14E-11
1.1400.93641606570.93641606566898694-2.97E-11
1.1450.93471115620.93471115622975964 2.05E-11
1.1500.93304093110.93304093109978414 6.51E-12
1.1550.93140502170.93140502165323868-3.93E-11
1.1600.92980306660.92980306664109957 4.51E-11
1.1650.92823471200.92823471196190366-2.59E-11
1.1700.92669961060.92669961062266581 2.10E-11
1.1750.92519742250.92519742251686099 1.24E-11
1.1800.92372781430.92372781430006712-1.17E-11
1.1850.92229045910.92229045925047382 1.49E-10
1.1900.92088503710.92088503713299241 2.60E-11
1.1950.91951123410.91951123406686597-2.98E-11
1.2000.91816874240.91816874239667101-1.67E-11
1.2050.91685726060.91685726056661909-3.28E-11
1.2100.91557649300.91557649299805532 8.85E-12
1.2150.91432614000.91432614997006778 9.98E-9
1.2200.91310594750.91310594750311536 1.37E-11
1.2250.91191560710.91191560725927312 1.49E-10
1.2300.91075485640.91075485637655895-1.50E-11
1.2350.90962342740.90962342744425173 4.03E-11
1.2400.90852105830.90852105834198582 4.21E-11
1.2450.90744749220.90744749215126341-5.77E-11
1.2500.90640247710.90640247705547716-3.68E-11
1.2550.90538576630.90538576624240463-5.23E-11
1.2600.90439711780.90439711780910215 2.01E-11
1.2650.90343629460.90343629466913566 5.78E-11
1.2700.90250306450.90250306446208062-5.13E-11
1.2750.90159719940.90159719946523187 5.66E-11
1.2800.90071847650.90071847650745973 5.78E-13
1.2850.89986667690.89986667689491762 5.55E-12
1.2900.89904158630.89904158628967101-3.93E-12
1.2950.89824299470.89824299468914737-1.72E-11
1.3000.89747069630.89747069630804477 2.65E-12
1.3050.89672448950.89672448951215833 2.37E-11
1.3100.89600417670.89600417674396082 4.53E-11
1.3150.89530956440.89530956444995535 5.43E-11
1.3200.89464046300.89464046300975775 1.28E-11
1.3250.89399668660.89399668666686083 7.95E-11
1.3300.89337805350.89337805346103716-3.97E-11
1.3350.89278438500.89278438516233538 1.51E-10
1.3400.89221550720.89221550720663356 1.43E-11
1.3450.89167124850.89167124863270442 1.24E-10
1.3500.89115144200.89115144202666452 3.78E-11
1.3550.89065592350.89065592343803057-5.12E-11
1.3600.89018453240.8901845323574008-5.70E-11
1.3650.88973711160.88973711163470881 3.11E-11
1.3700.88931350740.88931350742948501 4.09E-11
1.3750.88891356920.88891356915622532-5.89E-11
1.3800.88853714940.88853714943101736 2.03E-11
1.3850.88818410410.88818410401940351-9.53E-11
1.3900.88785429180.88785429178544073-1.00E-11
1.3950.88754757480.88754757464193323-1.49E-10
1.4000.88726381750.88726381750180738-7.13E-12
1.4050.88700288840.88700288823059736-1.66E-10
1.4100.88676465760.88676465760002188 3.66E-12
1.4150.88654899930.88654899924499497-4.45E-11
1.4200.88635578960.88635578960951567-1.60E-12
1.4250.88618490810.88618490791840432-1.81E-10
1.4300.88603623610.88603623612466142 2.35E-11
1.4350.88590965870.88590965887072826 1.59E-10
1.4400.88580506350.88580506344804788-5.45E-11
1.4450.88572233970.88572233975753722 5.12E-11
1.4500.88566138030.88566138027095553-3.63E-11
1.4550.88562207000.88562207999314335 9.99E-9
1.4600.88560433640.88560433642511449 3.29E-11
1.4650.88560804950.88560804952797856 4.00E-11
1.4700.88563312170.88563312168767672-2.25E-11
1.4750.88567945750.88567945767984679 1.68E-10
1.4800.88574696460.88574696463853297 3.58E-11
1.4850.88583555200.88583555202000774 1.39E-11
1.4900.88594513160.885945131572484-2.22E-11
1.4950.88607561740.88607561730422169-9.20E-11
1.5000.88622692550.88622692545275816-5.14E-11
1.5050.88639897440.88639897445482596 5.62E-11
1.5100.88659168500.88659168491694862-8.75E-11
1.5150.88680497970.88680497958669369-1.15E-10
1.5200.88703878330.88703878332457031 3.78E-11
1.5250.88729302310.88729302307655866-3.89E-11
1.5300.88756762780.88756762784725507 5.05E-11
1.5350.88786252870.88786252867361892-2.97E-11
1.5400.88817765860.88817765859552456-1.03E-11
1.5450.88851295270.88851295264558472-4.41E-11
1.5500.88886834780.88886834780261559 2.74E-12
1.5550.88924378300.88924378298210571-1.06E-11
1.5600.88963919900.88963919900923583-3.65E-12
1.5650.89005453870.89005453859597561-1.04E-10
1.5700.89048974630.89048974631869759 2.61E-11
1.5750.89094476860.89094476859629979 8.93E-12
1.5800.89141955370.89141955366882042-2.38E-11
1.5850.89191405150.8919140515765388 8.47E-11
1.5900.89242821410.8924282141395512 3.07E-11
1.5950.89296199490.89296199493781103 4.74E-11
1.6000.89351534930.89351534928506793-2.24E-11
1.6050.89408823420.89408823423580575 3.63E-11
1.6100.89468060850.89468060852796683 2.74E-11
1.6150.89529243270.89529243259029823-9.74E-11
1.6200.89592366850.89592366851824745 2.86E-11
1.6250.89657428000.89657428005659789 6.46E-11
1.6300.89724423260.89724423258250552-7.80E-12
1.6350.89793349300.89793349308892934 9.89E-11
1.6400.89864203020.89864203016845012-2.68E-11
1.6450.89936981380.89936981399746452 2.04E-10
1.6500.90011681630.9001168163207548 1.21E-11
1.6550.90088301040.90088301043641827 2.24E-11
1.6600.90166837120.90166837118115595-1.49E-11
1.6650.90247287480.90247287490643413 1.16E-10
1.6700.90329649950.9032964995021503-1.09E-11
1.6750.90413922430.90413922432675797 3.24E-11
1.6800.90500103020.90500103023115419 4.40E-11
1.6850.90588189960.90588189953639731-7.63E-11
1.6900.90678181600.90678181602099839 9.93E-12
1.6950.90770076500.90770076490852225-9.63E-11
1.7000.90863873290.90863873285549646-5.97E-11
1.7050.90959570790.90959570793962097 4.25E-11
1.7100.91057167960.9105716796482709 5.89E-11
1.7150.91156663900.91156663886729161-1.31E-10
1.7200.91258057790.91258057787007674-1.93E-11
1.7250.91361349040.91361349029479011-1.16E-10
1.7300.91466537120.91466537118231861-2.63E-11
1.7350.91573621710.9157362168940244-2.15E-10
1.7400.91682602520.91682602514979106-5.47E-11
1.7450.91793479500.91793479500653363 8.97E-12
1.7500.91906252680.91906252684888312 3.95E-11
1.7550.92020922240.92020922238011904-3.48E-11
1.7600.92137488460.92137488461334993 4.68E-12
1.7650.92255951780.92255951786293755 4.88E-11
1.7700.92376312770.9237631277361581 2.96E-11
1.7750.92498572110.92498572112510025 2.89E-11
1.7800.92622730620.92622730619879157 8.37E-12
1.7850.92748789260.92748789239555507-1.97E-10
1.7900.92876749040.92876749040057904-3.84E-12
1.7950.93006611230.93006611219852275-1.13E-10
1.8000.93138377100.93138377097715253-2.97E-11
1.8050.93272048110.93272048117993289 8.20E-11
1.8100.93407625850.93407625848467779-2.05E-11
1.8150.93545111980.93545111979719375 8.27E-12
1.8200.93684508320.93684508324512517 4.80E-11
1.8250.93825816820.93825816817200214-2.82E-11
1.8300.93969039510.93969039513148056 1.86E-11
1.8350.94114178590.94114178588178177-2.64E-11
1.8400.94261236340.94261236338031951-2.35E-11
1.8450.94410215190.94410215177851575-1.22E-10
1.8500.94561117640.94561117639912362-2.02E-12
1.8550.94713946370.94713946380190617 9.43E-11
1.8600.94868704170.94868704167359708-2.86E-11
1.8650.95025393890.95025393889348797-1.33E-11
1.8700.95184018550.95184018551169203 9.61E-12
1.8750.95344581270.95344581274503493 5.77E-11
1.8800.95507085300.95507085297311556-2.73E-11
1.8850.95671533980.95671533973453671-6.02E-11
1.8900.95837930770.95837930772329927 1.97E-11
1.8950.96006279270.960062792785362 8.60E-11
1.9000.96176583190.96176583191536336 2.60E-11
1.9050.96348846320.96348846325350124 5.75E-11
1.9100.96523072610.96523072608257054-3.05E-11
1.9150.96699266080.96699266080453206 5.78E-13
1.9200.96877430900.96877430902013406 1.66E-11
1.9250.97057571340.97057571340334281-3.67E-12
1.9300.97239691780.9723969177808085-5.87E-12
1.9350.97423796720.97423796710926569-8.59E-11
1.9400.97609890750.97609890747347727-2.67E-11
1.9450.97797978610.97797978608432246-2.76E-11
1.9500.97988065130.9798806512770295-3.65E-11
1.9550.98180155240.98180155250954815 1.02E-10
1.9600.98374254040.98374254036106346-5.01E-11
1.9650.98570366640.985703666530647 1.27E-10
1.9700.98768498380.98768498383604675 4.68E-11
1.9750.98968654620.98968654618919183-1.77E-11
1.9800.99170840870.99170840868869103-3.22E-12
1.9850.99375062740.9937506274792185 6.46E-11
1.9900.99581325980.99581325984380575 4.71E-11
1.9950.99789636430.99789636418011041-1.27E-10
+ + +

The Psi Function

+ +This table was constructed from the published values in the +Handbook of Mathematical Functions, by Milton Abramowitz +and Irene A. Stegun, by Dover (1965), pp 267-270. + +Axiom implements the polygamma function which allows for multiple +derivatives. The Psi function is a special case of the polygamma +function for zero derivatives. For the purpose of this table it +is defined as: +
+   Psi(x) == polygamma(0,x)
+
+ +The first column is the point where the Gamma function is evaluated. +The second column is the value reported in the Handbook. +The third column is the actual value computed by Axiom at the given point. +The fourth column is the difference of Axiom's value and the Handbook value. + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
pointHandbook ValueAxiom Computed ValueDifference
1.000-0.5772156649-0.57721566490153275-1.53E-12
1.005-0.5690209113-0.56902091134438304 -4.43E-11
1.010-0.5608854579-0.56088545786867472 3.13E-11
1.015-0.5528085156-0.55280851559434629 5.65E-12
1.020-0.5447893105-0.54478931045617984 4.38E-11
1.025-0.5368270828-0.53682708284938863 -4.93E-11
1.030-0.5289210873-0.5289210872854303 1.45E-11
1.035-0.5210705921-0.52107059205771 4.22E-11
1.040-0.5132748789-0.51327487891683021 -1.68E-11
1.045-0.5055332428-0.50553324275508449 4.49E-11
1.050-0.4978449913-0.49784499129987031 1.29E-13
1.055-0.4902094448-0.49020944481574569 -1.57E-11
1.060-0.4826259358-0.48262593581482538 -1.48E-11
1.065-0.4750938088-0.47509380877526647 2.47E-11
1.070-0.4676124199-0.46761241986755342 3.24E-11
1.075-0.4601811367-0.4601811366883593 1.16E-11
1.080-0.4527993380-0.45279933800171246 -1.71E-12
1.085-0.4454664135-0.44546641348725191 1.27E-11
1.090-0.4381817635-0.43818176349533489 4.66E-12
1.095-0.4309447988-0.43094479880878706 -8.78E-12
1.100-0.4237549404-0.42375494041107653 -1.10E-11
1.105-0.4166116193-0.41661161926071655 3.92E-11
1.110-0.4095142761-0.40951427607169383 2.83E-11
1.115-0.4024623611-0.40246236109974648 2.53E-13
1.120-0.3954553339-0.39545533393429283 -3.42E-11
1.125-0.3884926633-0.38849266329585463 4.14E-12
1.130-0.3815738268-0.38157382683879215 -3.87E-11
1.135-0.3746983110-0.37469831095919082 4.08E-11
1.140-0.3678656106-0.36786561060774969 -7.74E-12
1.145-0.3610752291-0.361075229107509 -7.50E-12
1.150-0.3543266780-0.35432667797627904 2.37E-11
1.155-0.3476194768-0.34761947675362337 4.63E-11
1.160-0.3409531528-0.34095315283226135 -3.22E-11
1.165-0.3343272413-0.3343272412937619 6.23E-12
1.170-0.3277412847-0.3277412847483927 -4.83E-11
1.175-0.3211948332-0.3211948331790081 2.09E-11
1.180-0.3146874438-0.31468744378886082 1.11E-11
1.185-0.3082186809-0.30821868085320625 4.67E-11
1.190-0.3017881156-0.30178811557461016 2.53E-11
1.195-0.2953953259-0.2953953259418296 -4.18E-11
1.200-0.2890398966-0.28903989659218843 7.81E-12
1.205-0.2827214187-0.28272141867731704 2.26E-11
1.210-0.2764394897-0.2764394897321919 -3.21E-11
1.215-0.2701937135-0.27019371354735244 -4.73E-11
1.220-0.2639837000-0.26398370004422023 -4.42E-11
1.225-0.2578090652-0.25780906515343338 4.65E-11
1.230-0.2516694307-0.25166943069609982 3.90E-12
1.235-0.2455644243-0.24556442426789726 3.21E-11
1.240-0.2394936791-0.23949367912593666 -2.59E-11
1.245-0.2334568341-0.23345683407831253 2.16E-11
1.250-0.2274535334-0.22745353337626528 2.37E-11
1.255-0.2214834266-0.22148342660888165 -8.88E-12
1.260-0.2155461686-0.21554616860026521 -2.65E-13
1.265-0.2096414193-0.20964141930911384 -9.11E-12
1.270-0.2037688437-0.20376884373062343 -3.06E-11
1.275-0.1979281118-0.19792811180067393 -6.73E-13
1.280-0.1921188983-0.19211889830222173 -2.22E-12
1.285-0.1863408828-0.18634088277384209 2.61E-11
1.290-0.1805937494-0.1805937494203691 -2.03E-11
1.295-0.1748771870-0.17487718702556942 -2.55E-11
1.300-0.1691908889-0.16919088886679934 3.32E-11
1.305-0.1635345526-0.163534552631597 -3.15E-11
1.310-0.1579078803-0.15790788033614178 -3.61E-11
1.315-0.1523105782-0.15231057824555994 -4.55E-11
1.320-0.1467423568-0.1467423567959959 4.00E-12
1.325-0.1412029305-0.14120293051842803 -1.84E-11
1.330-0.1356920180-0.13569201796416941 3.58E-11
1.335-0.1302093416-0.13020934163201769 -3.20E-11
1.340-0.1247546279-0.12475462789700376 2.99E-12
1.345-0.1193276069-0.11932760694070754 -4.07E-11
1.350-0.1139280127-0.11392801268308839 1.69E-11
1.355-0.1085555827-0.10855558271580501 -1.58E-11
1.360-0.1032100582-0.10321005823697738 -3.69E-11
1.365-0.0978911840-0.097891183987354968 1.26E-11
1.370-0.0925987082-0.092598708187860979 1.21E-11
1.375-0.0873323825-0.087332382478473081 2.15E-11
1.380-0.0820919619-0.082091961858406615 4.15E-11
1.385-0.0768772046-0.076877204627574525 -2.75E-11
1.390-0.0716878723-0.071687872329281643 -2.92E-11
1.395-0.0665237297-0.066523729694132228 5.86E-12
1.400-0.0613845446-0.061384544585116108 1.48E-11
1.405-0.0562700879-0.056270087943841696 -4.38E-11
1.410-0.0511801337-0.051180133737897426 -3.78E-11
1.415-0.0461144589-0.04.6114458909301992 -9.30E-12
1.420-0.0410728433-0.041072843324024277 -2.40E-11
1.425-0.0360550697-0.036055069722547906 -2.25E-11
1.430-0.0310609237-0.031060923671447194 2.85E-11
1.435-0.0260901935-0.02609019351596098 -1.59E-11
1.440-0.0211426703-0.021142670333530678 -3.35E-11
1.445-0.0162181479-0.016218147888283685 1.17E-11
1.450-0.0113164226-0.011316422586445718 1.35E-11
1.455-0.0064372934-0.0064372934326406561 -3.26E-11
1.460-0.0015805620-0.0015805619870833398 1.29E-11
1.4650.00325396770.0032539676763745362 -2.36E-11
1.4700.00806648900.0080664890113649745 1.13E-11
1.4750.01285719300.012857193039295334 3.92E-11
1.4800.01762626840.017626268388849287 -1.11E-11
1.4850.02237390130.022373901334705404 3.47E-11
1.4900.02710027580.027100275835486465 3.54E-11
1.4950.03180557360.031805573570971468 -2.90E-11
1.5000.03648997400.036489973978576673 -2.14E-11
1.5050.04115365430.041153654289123542 -1.08E-11
1.5100.04579678960.045796789561914686 -3.80E-11
1.5150.05041955270.050419552719128236 1.91E-11
1.5200.05502211460.055022114579551307 -2.04E-11
1.5250.05960464390.05960464389166209 -8.33E-12
1.5300.06416730740.064167307366077231 -3.39E-11
1.5350.06871026970.068710269707385141 7.38E-12
1.5400.07323369360.073233693645366138 4.53E-11
1.5450.07773773000.077737739965624497 9.96E-9
1.5500.08222256750.082222567539644631 3.96E-11
1.5550.08668833340.086688333354268288 -4.57E-11
1.5600.09113519250.091135192540635401 4.06E-11
1.5650.09556329840.095563298402570163 2.57E-12
1.5700.09997280240.099972802444444731 4.44E-11
1.5750.10436385440.10436385439851947 -1.48E-12
1.5800.10873660230.10873660225178161 -4.82E-11
1.5850.11309119230.11309119227228603 -2.77E-11
1.5900.11742776900.11742776903501095 3.50E-11
1.5950.12174647540.12174647544723916 4.72E-11
1.6000.12604745280.12604745277347584 -2.65E-11
1.6050.13033084070.13033084065991318 -4.00E-11
1.6100.13459677720.13459677715844587 -4.15E-11
1.6150.13884539880.13884539875025736 -4.97E-11
1.6200.14307684040.14307684036898005 -3.10E-11
1.6250.14729123540.14729123542343325 2.34E-11
1.6300.15148871580.15148871581995815 1.99E-11
1.6350.15566941200.15566941198435302 -1.56E-11
1.6400.15983345290.15983345288341522 -1.65E-11
1.6450.16398096600.16398096604610457 4.61E-11
1.6500.16811207760.16811207758432767 -1.56E-11
1.6550.17222691220.17222691221335784 1.33E-11
1.6600.17632559330.17632559327189457 -2.81E-11
1.6650.18040824270.18040824274177392 4.17E-11
1.6700.18447498130.1844749812673292 -3.26E-11
1.6750.18852592820.18852592817442249 -2.55E-11
1.6800.19256120150.19256120148913258 -1.08E-11
1.6850.19658091800.19658091795613342 -4.38E-11
1.6900.20058519310.20058519305674649 -4.32E-11
1.6950.20457414100.20457414102668603 2.66E-11
1.7000.20854787490.20854787487349435 -2.65E-11
1.7050.21250650640.21250650639368796 -6.31E-12
1.7100.21645014620.21645014618960501 -1.03E-11
1.7150.22037890370.2203789036859658 -1.40E-11
1.7200.22429288710.22429288714615725 4.61E-11
1.7250.22819220370.22819220368823745 -1.17E-11
1.7300.23207695930.23207695930067274 6.72E-13
1.7350.23594725890.23594725885781176 -4.21E-11
1.7400.23980320610.23980320613509676 3.50E-11
1.7450.24364490380.24364490382402559 2.40E-11
1.7500.24747245350.2474724535468612 4.68E-11
1.7550.25128595590.25128595587109781 -2.89E-11
1.7600.25508551030.25508551032368809 2.36E-11
1.7650.25887121540.25887121540503744 5.03E-12
1.7700.26264316860.26264316860276249 2.76E-12
1.7750.26640146640.2664014664052331 5.23E-12
1.7800.27014620430.27014620431488368 1.48E-11
1.7850.27387747690.27387747686131236 -3.86E-11
1.7900.27759537760.27759537761416786 1.41E-11
1.7950.28129999920.2812999991958266 -4.17E-12
1.8000.28499143330.2849914332938619 -6.13E-12
1.8050.28866977070.28866977067331689 -2.66E-11
1.8100.29233510120.29233510118877948 -1.12E-11
1.8150.29598751380.29598751379626109 -3.73E-12
1.8200.29962709660.29962709656488773 -3.51E-11
1.8250.30325393670.30325393668840539 -1.15E-11
1.8300.30686812050.30686812049650136 -3.49E-12
1.8350.31046973350.31046973346594764 -3.40E-11
1.8400.31405886020.31405886023156859 3.15E-11
1.8450.31763558460.31763558459703256 -2.96E-12
1.8500.32119998950.32119998954547946 4.54E-11
1.8550.32475215720.32475215724997797 4.99E-11
1.8600.32829216910.32829216908382075 -1.61E-11
1.8650.33182010560.33182010563065989 3.06E-11
1.8700.33533604670.33533604669448569 -5.51E-12
1.8750.33884007130.33884007130944738 9.44E-12
1.8800.34233225770.34233225774952925 4.95E-11
1.8850.34581268350.34581268353806771 3.80E-11
1.8900.34928142550.34928142545713492 -4.28E-11
1.8950.35273855960.35273855955676792 -4.32E-11
1.9000.35618416120.35618416116406026 -3.59E-11
1.9050.35961830490.35961830489211799 -7.88E-12
1.9100.36304106460.36304106464888108 4.88E-11
1.9150.36645251360.36645251364580167 4.58E-11
1.9200.36985272440.36985272440640171 6.40E-12
1.9250.37324176880.37324176877469795 -2.53E-11
1.9300.37661971790.37661971792349891 2.34E-11
1.9350.37998664240.37998664236258128 -3.74E-11
1.9400.38334261190.38334261194674013 4.67E-11
1.9450.38668769590.38668769588372298 -1.62E-11
1.9500.39002196270.39002196274204304 4.20E-11
1.9550.39334548050.39334548045868012 -4.13E-11
1.9600.39665831630.39665831634666171 4.66E-11
1.9650.39996053710.39996053710254509 2.54E-12
1.9700.40325220880.40325220881377177 1.37E-11
1.9750.40653339700.40653339696592627 -3.40E-11
1.9800.40980416640.40980416644989071 4.98E-11
1.9850.41306458160.41306458156888626 -3.11E-11
1.9900.41631470600.41631470604541487 4.54E-11
1.9950.41955460300.41955460302810832 2.81E-11
2.0000.42278433510.42278433509846725 -1.53E-12
+<> +@ + \subsection{draw.xhtml} <>= <> @@ -21613,6 +42463,10 @@ infinity; the step size is any positive integer. Mathematical Methods MIT 18-08 Mathematical Methods for Engineers Course Notes + + CATS + Computer Algebra Test Suite + <> @ @@ -22148,7 +43002,7 @@ static char axiom_bits[] = { 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00}; @ - + \section{License} <>= --Copyright (c) 2007 Arthur C. Ralfs