psi function

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1: 5.2 Definitions

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§5.2(i) Gamma and Psi Functions

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Euler’s Integral

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5.2.1 $$\mathrm{\Gamma }\left(z\right)={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{z-1}dt,$$ $\mathrm{\Re }z>0$.

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Defines:

$\mathrm{\Gamma }\left(z\right)$: gamma function

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

6.1.1

Referenced by:

§10.43(iii), (25.11.27), (25.11.28), (25.5.6), (25.5.7), §5.9(i), §5.9(ii), §8.21(ii), (9.12.17)

Permalink:

http://dlmf.nist.gov/5.2.E1

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Annotations for §5.2(i), §5.2(i), §5.2 and Ch.5

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5.2.2 $$\psi \left(z\right)={\mathrm{\Gamma }}^{\prime }\left(z\right)/\mathrm{\Gamma }\left(z\right),$$ $z\ne 0,-1,-2,\mathrm{\dots }$.

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Defines:

$\psi \left(z\right)$: psi (or digamma) function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function and $z$: complex variable

A&S Ref:

6.3.1

Permalink:

http://dlmf.nist.gov/5.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.2(i), §5.2(i), §5.2 and Ch.5

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2: 5.1 Special Notation

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$j,m,n$	nonnegative integers.
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►The main functions treated in this chapter are the gamma function $\mathrm{\Gamma }\left(z\right)$, the psi function (or digamma function) $\psi \left(z\right)$, the beta function $\mathrm{B}(a,b)$, and the $q$-gamma function ${\mathrm{\Gamma }}_q\left(z\right)$. … ►Alternative notations for the psi function are: $\mathrm{\Psi }(z-1)$ (Gauss) Jahnke and Emde (1945); $\mathrm{\Psi }(z)$ Davis (1933); $𝖥(z-1)$ Pairman (1919).

3: 5.21 Methods of Computation

… ►For the left half-plane we can continue the backward recurrence or make use of the reflection formula (5.5.3). ►Similarly for $\ln \mathrm{\Gamma }\left(z\right)$, $\psi \left(z\right)$, and the polygamma functions. …

4: 5.23 Approximations

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§5.23(i) Rational Approximations

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§5.23(ii) Expansions in Chebyshev Series

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§5.23(iii) Approximations in the Complex Plane

►See Schmelzer and Trefethen (2007) for a survey of rational approximations to various scaled versions of $\mathrm{\Gamma }\left(z\right)$. …

5: 5.16 Sums

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5.16.1 $$\sum _{k=1}^{\mathrm{\infty }}{(-1)}^k{\psi }^{\prime }\left(k\right)=-\frac{{\pi }^2}{8},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/5.16.E1

Encodings:

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Annotations for §5.16 and Ch.5

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5.16.2 $$\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k}{\psi }^{\prime }\left(k+1\right)=\zeta \left(3\right)=-\frac{1}{2}{\psi }^{\prime \prime }\left(1\right).$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\psi \left(z\right)$: psi (or digamma) function and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/5.16.E2

Encodings:

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Annotations for §5.16 and Ch.5

►For further sums involving the psi function see Hansen (1975, pp. 360–367). …

6: 5.22 Tables

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§5.22(ii) Real Variables

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§5.22(iii) Complex Variables

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7: 5.3 Graphics

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Figure 5.3.3: $\psi \left(x\right)$. Magnify

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Figure 5.3.6: $|\psi \left(x+\mathrm{i}y\right)|$. Magnify 3D Help

8: 5.5 Functional Relations

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5.5.1 $$\mathrm{\Gamma }\left(z+1\right)=z\mathrm{\Gamma }\left(z\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function and $z$: complex variable

A&S Ref:

6.1.15

Referenced by:

§15.4(ii), §15.4(iii), (25.11.27), (25.5.6), §5.4(i)

Permalink:

http://dlmf.nist.gov/5.5.E1

Encodings:

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Annotations for §5.5(i), §5.5 and Ch.5

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5.5.2 $$\psi \left(z+1\right)=\psi \left(z\right)+\frac{1}{z}.$$

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Symbols:

$\psi \left(z\right)$: psi (or digamma) function and $z$: complex variable

A&S Ref:

6.3.5

Permalink:

http://dlmf.nist.gov/5.5.E2

Encodings:

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Annotations for §5.5(i), §5.5 and Ch.5

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5.5.3 $$\mathrm{\Gamma }\left(z\right)\mathrm{\Gamma }\left(1-z\right)=\pi /\sin \left(\pi z\right),$$ $z\ne 0,\pm 1,\mathrm{\dots }$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function and $z$: complex variable

A&S Ref:

6.1.17 (without the condition on $z$.)

Referenced by:

§10.19(i), §15.4(ii), §15.8(ii), (25.9.2), §5.21, (9.10.18), (9.12.15)

Permalink:

http://dlmf.nist.gov/5.5.E3

Encodings:

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Annotations for §5.5(ii), §5.5 and Ch.5

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5.5.7 $$\prod _{k=1}^{n-1}\mathrm{\Gamma }\left(\frac{k}{n}\right)={(2\pi )}^{(n-1)/2}{n}^{-1/2}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $n$: nonnegative integer and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/5.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §5.5(iii), §5.5(iii), §5.5 and Ch.5

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5.5.8 $$\psi \left(2z\right)=\frac{1}{2}\left(\psi \left(z\right)+\psi \left(z+\frac{1}{2}\right)\right)+\ln 2,$$

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Symbols:

$\psi \left(z\right)$: psi (or digamma) function, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

6.3.8

Referenced by:

§5.5(iii)

Permalink:

http://dlmf.nist.gov/5.5.E8

Encodings:

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Annotations for §5.5(iii), §5.5(iii), §5.5 and Ch.5

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9: 5.7 Series Expansions

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

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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:

pMML, png, TeX

See also:

Annotations for §5.7(i), §5.7 and Ch.5

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5.7.4 $$\psi \left(1+z\right)=-\gamma +\sum _{k=2}^{\mathrm{\infty }}{(-1)}^k\zeta \left(k\right)z^{k-1},$$ ,

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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:

pMML, png, TeX

See also:

Annotations for §5.7(i), §5.7 and Ch.5

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5.7.5 $$\psi \left(1+z\right)=\frac{1}{2z}-\frac{\pi }{2}\cot \left(\pi z\right)+\frac{1}{z^2-1}+1-\gamma -\sum _{k=1}^{\mathrm{\infty }}(\zeta \left(2k+1\right)-1)z^{2k},$$ , $z\ne 0,\pm 1$.

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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:

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See also:

Annotations for §5.7(i), §5.7 and Ch.5

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§5.7(ii) Other Series

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5.7.7 $$\psi \left(\frac{z+1}{2}\right)-\psi \left(\frac{z}{2}\right)=2\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k}{k+z}.$$

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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:

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See also:

Annotations for §5.7(ii), §5.7 and Ch.5

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10: 5.4 Special Values and Extrema

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§5.4(ii) Psi Function

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5.4.14 $$\psi \left(n+1\right)=\sum _{k=1}^n\frac{1}{k}-\gamma ,$$

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Symbols:

$\gamma$: Euler’s constant, $\psi \left(z\right)$: psi (or digamma) function, $n$: nonnegative integer and $k$: nonnegative integer

A&S Ref:

6.3.2

Permalink:

http://dlmf.nist.gov/5.4.E14

Encodings:

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See also:

Annotations for §5.4(ii), §5.4 and Ch.5

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5.4.19 $$\psi \left(\frac{p}{q}\right)=-\gamma -\ln q-\frac{\pi }{2}\cot \left(\frac{\pi p}{q}\right)+\frac{1}{2}\sum _{k=1}^{q-1}\cos \left(\frac{2\pi kp}{q}\right)\ln \left(2-2\cos \left(\frac{2\pi k}{q}\right)\right).$$

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Symbols:

$\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\cot z$: cotangent function, $\psi \left(z\right)$: psi (or digamma) function, $\ln z$: principal branch of logarithm function, $q$: real or complex variable and $k$: nonnegative integer

Referenced by:

§5.19(i), §5.4(ii)

Permalink:

http://dlmf.nist.gov/5.4.E19

Encodings:

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See also:

Annotations for §5.4(ii), §5.4 and Ch.5

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§5.4(iii) Extrema

… ►As $n\to \mathrm{\infty }$, …