5 Gamma Function Properties 5.6 Inequalities 5.8 Infinite Products

§5.7 Series Expansions

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Permalink:: http://dlmf.nist.gov/5.7
See also:: Annotations for Ch.5

§5.7(i) Maclaurin and Taylor Series
§5.7(ii) Other Series

§5.7(i) Maclaurin and Taylor Series

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Keywords:: Maclaurin series, Taylor series, gamma function, logarithm, psi function, reciprocal
Notes:: For (5.7.1)–(5.7.2) see Wrench (1968) (errors on p. 621 are corrected here). For (5.7.3)–(5.7.5) see Erdélyi et al. (1953a, pp. 45–47).
Permalink:: http://dlmf.nist.gov/5.7.i
See also:: Annotations for §5.7 and Ch.5

Throughout this subsection $\zeta\left(k\right)$ is as in Chapter 25.

5.7.1 $\frac{1}{\Gamma\left(z\right)}=\sum_{k=1}^{\infty}c_{k}z^{k},$

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, : nonnegative integer, : complex variable and $c_{k}$ : coefficient
A&S Ref:: 6.1.34
Referenced by:: §5.7(i)
Permalink:: http://dlmf.nist.gov/5.7.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §5.7(i), §5.7 and Ch.5

where $c_{1}=1$ , $c_{2}=\gamma$ , and

5.7.2 $(k-1)c_{k}=\gamma c_{k-1}-\zeta\left(2\right)c_{k-2}+\zeta\left(3\right)c_{k-3% }-\dots+(-1)^{k}\zeta\left(k-1\right)c_{1},$ $k\geq 3$ .

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Symbols:: $\gamma$ : Euler’s constant, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, : nonnegative integer and $c_{k}$ : coefficient
Referenced by:: §5.7(i)
Permalink:: http://dlmf.nist.gov/5.7.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §5.7(i), §5.7 and Ch.5

For 15D numerical values of $c_{k}$ see Abramowitz and Stegun (1964, p. 256), and for 31D values see Wrench (1968).

5.7.3 $\ln\Gamma\left(1+z\right)=-\ln\left(1+z\right)+z(1-\gamma)+\sum_{k=2}^{\infty}% (-1)^{k}(\zeta\left(k\right)-1)\frac{z^{k}}{k},$

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\gamma$ : Euler’s constant, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\ln\NVar{z}$ : principal branch of logarithm function, : nonnegative integer and : complex variable
A&S Ref:: 6.1.33
Referenced by:: §5.21, §5.7(i)
Permalink:: http://dlmf.nist.gov/5.7.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §5.7(i), §5.7 and Ch.5

5.7.4 $\psi\left(1+z\right)=-\gamma+\sum_{k=2}^{\infty}(-1)^{k}\zeta\left(k\right)z^{% k-1},$

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Symbols:: $\gamma$ : Euler’s constant, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, : nonnegative integer and : complex variable
A&S Ref:: 6.3.14
Permalink:: http://dlmf.nist.gov/5.7.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §5.7(i), §5.7 and Ch.5

5.7.5 $\psi\left(1+z\right)=\frac{1}{2z}-\frac{\pi}{2}\cot\left(\pi z\right)+\frac{1}% {z^{2}-1}+1-\gamma-\sum_{k=1}^{\infty}(\zeta\left(2k+1\right)-1)z^{2k},$

, $z\neq 0,\pm 1$ .

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Symbols:: $\gamma$ : Euler’s constant, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cot\NVar{z}$ : cotangent function, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, : nonnegative integer and : complex variable
A&S Ref:: 6.3.15
Referenced by:: §5.7(i)
Permalink:: http://dlmf.nist.gov/5.7.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §5.7(i), §5.7 and Ch.5

For 20D numerical values of the coefficients of the Maclaurin series for $\Gamma\left(z+3\right)$ see Luke (1969b, p. 299).

§5.7(ii) Other Series

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Keywords:: expansions in partial fractions, psi function
Notes:: See Olver (1997b, p. 39) and Temme (1996b, pp. 54, 56).
Referenced by:: §5.19(i)
Permalink:: http://dlmf.nist.gov/5.7.ii
See also:: Annotations for §5.7 and Ch.5

When $z\neq 0,-1,-2,\dots$ ,

5.7.6 $\psi\left(z\right)=-\gamma-\frac{1}{z}+\sum_{k=1}^{\infty}\frac{z}{k(k+z)}=-% \gamma+\sum_{k=0}^{\infty}\left(\frac{1}{k+1}-\frac{1}{k+z}\right),$

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