§ 5.7. Series Expansions
Contents
§ 5.7(i). Maclaurin and Taylor Series
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- 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 is as in Chapter 25.
5.7.1
![\frac{1}{\Gamma\!\left(z\right)}=\sum _{{k=1}}^{\infty}c_{k}z^{k},](../../5/7/E1.png)
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- Defines:
-
: coefficient
- Symbols:
-
: Gamma function,
: nonnegative integer and
: complex variable
- A&S Ref:
- 6.1.34
- Referenced by:
- §5.7(i)
- Permalink:
- http://dlmf.nist.gov/5.7.E1
- Encodings:
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where ,
, 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},](../../5/7/E2.png)
![k\ge 3](../../5/7/m6.png)
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- Defines:
-
: coefficient
- Symbols:
-
: Euler's constant and
: nonnegative integer
- Referenced by:
- §5.7(i)
- Permalink:
- http://dlmf.nist.gov/5.7.E2
- Encodings:
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For 15D numerical values of 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},](../../5/7/E3.png)
![|z|<2](../../5/7/m4.png)
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- Symbols:
-
: Gamma function,
: Euler's constant,
: nonnegative integer and
: complex variable
- A&S Ref:
- 6.1.33
- Referenced by:
- §5.21, §5.7(i)
- Permalink:
- http://dlmf.nist.gov/5.7.E3
- Encodings:
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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}},](../../5/7/E4.png)
![|z|<1](../../5/7/m8.png)
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- Symbols:
-
: Euler's constant,
: Psi or digamma function,
: nonnegative integer and
: complex variable
- A&S Ref:
- 6.3.14
- Permalink:
- http://dlmf.nist.gov/5.7.E4
- Encodings:
- TeX, pMathML, png
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}},](../../5/7/E5.png)
![|z|<2](../../5/7/m4.png)
![z\neq 0,\pm 1](../../5/7/m9.png)
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- Symbols:
-
: Euler's constant,
: Psi or digamma function,
: nonnegative integer and
: complex variable
- A&S Ref:
- 6.3.15
- Referenced by:
- §5.7(i)
- Permalink:
- http://dlmf.nist.gov/5.7.E5
- Encodings:
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For 20D numerical values of the coefficients of the Maclaurin series for
see Luke (1969, p. 299).
§ 5.7(ii). Other Series
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- 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 ,
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),](../../5/7/E6.png)
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- Symbols:
-
: Euler's constant,
: Psi or digamma function,
: nonnegative integer and
: complex variable
- Referenced by:
- §5.19(i)
- Permalink:
- http://dlmf.nist.gov/5.7.E6
- Encodings:
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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}.](../../5/7/E7.png)
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- Symbols:
-
: Psi or digamma function,
: nonnegative integer and
: complex variable
- Permalink:
- http://dlmf.nist.gov/5.7.E7
- Encodings:
- TeX, pMathML, png
Also,
5.7.8
![\imagpart{\psi\!\left(iy+1\right)}=\sum _{{k=1}}^{\infty}\frac{y}{k^{2}+y^{2}}.](../../5/7/E8.png)
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- Symbols:
-
: Psi or digamma function,
: nonnegative integer and
: real variable
- A&S Ref:
- 6.3.13
- Permalink:
- http://dlmf.nist.gov/5.7.E8
- Encodings:
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