# §5.7 Series Expansions

## §5.7(i) Maclaurin and Taylor Series

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 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 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 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 5 5.7.5 $\displaystyle\psi\left(1+z\right)$ $\displaystyle=\frac{1}{2z}-\frac{\pi}{2}\cot\left(\pi z\right)+\frac{1}{z^{2}-% 1}+1-\gamma-\sum_{k=1}^{\infty}(\zeta\left(2k+1\right)-1)z^{2k},$ $|z|<2$, $z\neq 0,\pm 1$.

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

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 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 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, $k$: 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 5