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

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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		$\frac{1}{\Gamma\left(z\right)}=\sum_{k=1}^{\infty}c_{k}z^{k},$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $k$ : nonnegative integer, $z$ : 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		$(k-1)c_{k}=\gamma c_{k-1}-\zeta\left(2\right)c_{k-2}+\zeta\left(3\right)c_{k-3% }-\dots+(-1)^{k}\zeta\left(k-1\right)c_{1},$
$k\geq 3$ .
ⓘ Symbols: $\gamma$ : Euler’s constant, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $k$ : 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	$\displaystyle\ln\Gamma\left(1+z\right)$	$\displaystyle=-\ln\left(1+z\right)+z(1-\gamma)+\sum_{k=2}^{\infty}(-1)^{k}(% \zeta\left(k\right)-1)\frac{z^{k}}{k},$
$\|z\|<2$ .
ⓘ 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, $k$ : nonnegative integer and $z$ : 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	$\displaystyle\psi\left(1+z\right)$	$\displaystyle=-\gamma+\sum_{k=2}^{\infty}(-1)^{k}\zeta\left(k\right)z^{k-1},$
$\|z\|<1$ ,
ⓘ Symbols: $\gamma$ : Euler’s constant, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $k$ : nonnegative integer and $z$ : 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	$\displaystyle\psi\left(1+z\right)$	$\displaystyle=\frac{1}{2z}-\frac{\pi}{2}\cot\left(\pi z\right)+\frac{1}{z^{2}-% 1}+1-\gamma-\sum_{k=1}^{\infty}(\zeta\left(2k+1\right)-1)z^{2k},$
$\|z\|<2$ , $z\neq 0,\pm 1$ .
ⓘ 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, $k$ : nonnegative integer and $z$ : 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		$\psi\left(z\right)=-\gamma-\frac{1}{z}+\sum_{k=1}^{\infty}\frac{z}{k(k+z)}=-% \gamma+\sum_{k=0}^{\infty}\left(\frac{1}{k+1}-\frac{1}{k+z}\right),$
ⓘ Symbols: $\gamma$ : Euler’s constant, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $k$ : nonnegative integer and $z$ : complex variable Referenced by: §5.19(i) Permalink: http://dlmf.nist.gov/5.7.E6 Encodings: TeX, pMML, png See also: Annotations for §5.7(ii), §5.7 and Ch.5

and

5.7.7		$\psi\left(\frac{z+1}{2}\right)-\psi\left(\frac{z}{2}\right)=2\sum_{k=0}^{% \infty}\frac{(-1)^{k}}{k+z}.$
ⓘ Symbols: $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $k$ : nonnegative integer and $z$ : complex variable Referenced by: §15.4(iii) Permalink: http://dlmf.nist.gov/5.7.E7 Encodings: TeX, pMML, png See also: Annotations for §5.7(ii), §5.7 and Ch.5

Also,

5.7.8		$\Im\psi\left(1+\mathrm{i}y\right)=\sum_{k=1}^{\infty}\frac{y}{k^{2}+y^{2}}.$
ⓘ Symbols: $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\Im$ : imaginary part, $\mathrm{i}$ : imaginary unit, $k$ : nonnegative integer and $y$ : real variable A&S Ref: 6.3.13 Permalink: http://dlmf.nist.gov/5.7.E8 Encodings: TeX, pMML, png See also: Annotations for §5.7(ii), §5.7 and Ch.5

5.6 Inequalities 5.8 Infinite Products

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§5.7 Series Expansions

Contents

§5.7(i) Maclaurin and Taylor Series

§5.7(ii) Other Series