§ 5.7. Series Expansions

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

§ 5.7(i). Maclaurin and Taylor Series

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. (1953, pp. 45–47).
Keywords:: gamma function, psi function, Taylor series
Permalink:: http://dlmf.nist.gov/5.7.SS1

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

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

Defines:: $c_{k}$ : coefficient
Symbols:: $\Gamma\!\left(z\right)$ : Gamma function, $k$ : nonnegative integer and $z$ : complex variable
A&S Ref:: 6.1.34
Referenced by:: §5.7(i)
Permalink:: http://dlmf.nist.gov/5.7.E1
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where $c_{1}=1$ , $c_{2}=\EulerConstant$ , and

5.7.2 $(k-1)c_{k}=\EulerConstant c_{{k-1}}-\zeta\!\left(2\right)c_{{k-2}}+\zeta\!\left(3\right)c_{{k-3}}-\dots+(-1)^{k}\zeta\!\left(k-1\right)c_{1},$ $k\ge 3$ .

Defines:: $c_{k}$ : coefficient
Symbols:: $\EulerConstant$ : Euler's constant and $k$ : nonnegative integer
Referenced by:: §5.7(i)
Permalink:: http://dlmf.nist.gov/5.7.E2
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For 15D numerical values of $c_{k}$ see Abramowitz and Stegun (1964, p. 256), and for 31D values see Wrench (1968).

5.7.3 $\ln\Gamma\!\left(1+z\right)=-\ln\!\left(1+z\right)+z(1-\EulerConstant)+\sum _{{k=2}}^{\infty}(-1)^{k}(\zeta\!\left(k\right)-1)\frac{z^{k}}{k},$ $|z|<2$ .

Symbols:: $\Gamma\!\left(z\right)$ : Gamma function, $\EulerConstant$ : Euler's constant, $k$ : 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
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5.7.4 $\psi\!\left(1+z\right)=-\EulerConstant+\sum _{{k=2}}^{\infty}(-1)^{k}\zeta\!\left(k\right)z^{{k-1}},$ $|z|<1$ ,

Symbols:: $\EulerConstant$ : Euler's constant, $\psi\!\left(z\right)$ : 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
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5.7.5 $\psi\!\left(1+z\right)=\frac{1}{2z}-\frac{\pi}{2}\cot\!\left(\pi z\right)+\frac{1}{z^{2}-1}+1-\EulerConstant-\sum _{{k=1}}^{\infty}(\zeta\!\left(2k+1\right)-1)z^{{2k}},$ $|z|<2$ , $z\neq 0,\pm 1$ .

Symbols:: $\EulerConstant$ : Euler's constant, $\psi\!\left(z\right)$ : 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
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For 20D numerical values of the coefficients of the Maclaurin series for $\Gamma\!\left(z+3\right)$ see Luke (1969, p. 299).

§ 5.7(ii). Other Series

Notes:: See Olver (1997b, p. 39) and Temme (1996, pp. 54, 56).
Keywords:: psi function
Referenced by:: §5.19(i)
Permalink:: http://dlmf.nist.gov/5.7.SS2

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

5.7.6 $\psi\!\left(z\right)=-\EulerConstant-\frac{1}{z}+\sum _{{k=1}}^{\infty}\frac{z}{k(k+z)}=-\EulerConstant+\sum _{{k=0}}^{\infty}\left(\frac{1}{k+1}-\frac{1}{k+z}\right),$

Symbols:: $\EulerConstant$ : 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
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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:: $\psi\!\left(z\right)$ : Psi or digamma function, $k$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/5.7.E7
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Also,

5.7.8 $\imagpart{\psi\!\left(iy+1\right)}=\sum _{{k=1}}^{\infty}\frac{y}{k^{2}+y^{2}}.$

Symbols:: $\psi\!\left(z\right)$ : Psi or digamma function, $k$ : nonnegative integer and $y$ : real variable
A&S Ref:: 6.3.13
Permalink:: http://dlmf.nist.gov/5.7.E8
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§ 5.7. Series Expansions

Contents

§ 5.7(i). Maclaurin and Taylor Series

§ 5.7(ii). Other Series