§5.7 Series Expansions

Permalink:: http://dlmf.nist.gov/5.7

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

Throughout this subsection $\mathop{\zeta\/}\nolimits\!\left(k\right)$ is as in Chapter 25.

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : 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
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where $c_{1}=1$ , $c_{2}=\EulerConstant$ , and

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

Symbols:: $\EulerConstant$ : Euler’s constant, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $k$ : nonnegative integer and $c_{k}$ : coefficient
Referenced by:: §5.7(i)
Permalink:: http://dlmf.nist.gov/5.7.E2
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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 $\mathop{\ln\/}\nolimits\mathop{\Gamma\/}\nolimits\!\left(1+z\right)=-\mathop{\ln\/}\nolimits\!\left(1+z\right)+z(1-\EulerConstant)+\sum _{{k=2}}^{\infty}(-1)^{k}(\mathop{\zeta\/}\nolimits\!\left(k\right)-1)\frac{z^{k}}{k},$ $|z|<2$ .

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

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

Symbols:: $\EulerConstant$ : Euler’s constant, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\mathop{\cot\/}\nolimits z$ : cotangent function, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : 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 $\mathop{\Gamma\/}\nolimits\!\left(z+3\right)$ see Luke (1969b, p. 299).

§5.7(ii) Other Series

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

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

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

Symbols:: $\EulerConstant$ : Euler’s constant, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : 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 $\mathop{\psi\/}\nolimits\!\left(\frac{z+1}{2}\right)-\mathop{\psi\/}\nolimits\!\left(\frac{z}{2}\right)=2\sum _{{k=0}}^{\infty}\frac{(-1)^{k}}{k+z}.$

Symbols:: $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $k$ : nonnegative integer and $z$ : complex variable
Referenced by:: §15.4(iii)
Permalink:: http://dlmf.nist.gov/5.7.E7
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Also,

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

Symbols:: $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $\imagpart{}$ : imaginary part, $k$ : 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