§5.7 Series Expansions

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

5.7.1		$$\frac{1}{\mathrm{\Gamma }\left(z\right)}=\sum _{k=1}^{\mathrm{\infty }}{c}_k{z}^k,$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: 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		$$(k-1)c_k=\gamma c_{k-1}-\zeta \left(2\right)c_{k-2}+\zeta \left(3\right)c_{k-3}-\mathrm{\cdots }+{(-1)}^k\zeta \left(k-1\right)c_1,$$
$k\ge 3$.
ⓘ Symbols: $\gamma$: Euler’s constant, $\zeta \left(s\right)$: 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		$\ln \mathrm{\Gamma }\left(1+z\right)$	$=-\ln \left(1+z\right)+z(1-\gamma )+{\displaystyle \sum _{k=2}^{\mathrm{\infty }}}{(-1)}^k(\zeta \left(k\right)-1){\displaystyle \frac{z^k}{k}},$
.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, $\ln z$: 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		$\psi \left(1+z\right)$	$=-\gamma +{\displaystyle \sum _{k=2}^{\mathrm{\infty }}}{(-1)}^k\zeta \left(k\right)z^{k-1},$
,
ⓘ Symbols: $\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, $\psi \left(z\right)$: 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		$\psi \left(1+z\right)$	$={\displaystyle \frac{1}{2z}}-{\displaystyle \frac{\pi }{2}}\cot \left(\pi z\right)+{\displaystyle \frac{1}{z^2-1}}+1-\gamma -{\displaystyle \sum _{k=1}^{\mathrm{\infty }}}(\zeta \left(2k+1\right)-1)z^{2k},$
, $z\ne 0,\pm 1$.
ⓘ Symbols: $\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\psi \left(z\right)$: 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 $\mathrm{\Gamma }\left(z+3\right)$ see Luke (1969b, p. 299).

When $z\ne 0,-1,-2,\mathrm{\dots }$,

5.7.6		$$\psi \left(z\right)=-\gamma -\frac{1}{z}+\sum _{k=1}^{\mathrm{\infty }}\frac{z}{k(k+z)}=-\gamma +\sum _{k=0}^{\mathrm{\infty }}\left(\frac{1}{k+1}-\frac{1}{k+z}\right),$$
ⓘ Symbols: $\gamma$: Euler’s constant, $\psi \left(z\right)$: 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		$$\psi \left(\frac{z+1}{2}\right)-\psi \left(\frac{z}{2}\right)=2\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k}{k+z}.$$
ⓘ Symbols: $\psi \left(z\right)$: 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		$$\mathrm{\Im }\psi \left(1+\mathrm{i}y\right)=\sum _{k=1}^{\mathrm{\infty }}\frac{y}{k^2+y^2}.$$
ⓘ Symbols: $\psi \left(z\right)$: psi (or digamma) function, $\mathrm{\Im }$: 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