digamma function

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1: 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:

pMML, png, TeX

See also:

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:

pMML, png, TeX

See also:

Annotations for §5.16 and Ch.5

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2: 5.1 Special Notation

… ►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: 5.15 Polygamma Functions

… ►In particular, ${\psi }^{\prime }\left(z\right)$ is the trigamma function; ${\psi }^{\prime \prime }$, ${\psi }^{(3)}$, ${\psi }^{(4)}$ are the tetra-, penta-, and hexagamma functions respectively. … ►

5.15.1 $${\psi }^{\prime }\left(z\right)=\sum _{k=0}^{\mathrm{\infty }}\frac{1}{{(k+z)}^2},$$ $z\ne 0,-1,-2,\mathrm{\dots }$,

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Symbols:

$\psi \left(z\right)$: psi (or digamma) function, $k$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/5.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §5.15 and Ch.5

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5.15.5 $${\psi }^{(n)}\left(z+1\right)={\psi }^{(n)}\left(z\right)+{(-1)}^{n}n!z^{-n-1},$$

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Symbols:

$\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $n$: nonnegative integer and $z$: complex variable

A&S Ref:

6.4.6

Permalink:

http://dlmf.nist.gov/5.15.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §5.15 and Ch.5

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5.15.6 $${\psi }^{(n)}\left(1-z\right)+{(-1)}^{n-1}{\psi }^{(n)}\left(z\right)={(-1)}^n\pi \frac{d^n}{{dz}^n}\cot \left(\pi z\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\psi \left(z\right)$: psi (or digamma) function, $n$: nonnegative integer and $z$: complex variable

A&S Ref:

6.4.7

Permalink:

http://dlmf.nist.gov/5.15.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §5.15 and Ch.5

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5.15.7 $${\psi }^{(n)}\left(mz\right)=\frac{1}{m^{n+1}}\sum _{k=0}^{m-1}{\psi }^{(n)}\left(z+\frac{k}{m}\right).$$

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Symbols:

$\psi \left(z\right)$: psi (or digamma) function, $m$: nonnegative integer, $n$: nonnegative integer, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

6.4.8

Referenced by:

§5.15

Permalink:

http://dlmf.nist.gov/5.15.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §5.15 and Ch.5

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4: 5.7 Series Expansions

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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:

pMML, png, TeX

See also:

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

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

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Symbols:

$\gamma$: 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

Encodings:

pMML, png, TeX

See also:

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

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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:

pMML, png, TeX

See also:

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

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5.7.8 $$\mathrm{\Im }\psi \left(1+\mathrm{i}y\right)=\sum _{k=1}^{\mathrm{\infty }}\frac{y}{k^2+y^2}.$$

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Symbols:

$\psi \left(z\right)$: psi (or digamma) function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer and $y$: real variable

A&S Ref:

6.3.13

Permalink:

http://dlmf.nist.gov/5.7.E8

Encodings:

pMML, png, TeX

See also:

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

5: 5.4 Special Values and Extrema

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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:

pMML, png, TeX

See also:

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

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5.4.15 $$\psi \left(n+\frac{1}{2}\right)=-\gamma -2\ln 2+2\left(1+\frac{1}{3}+\mathrm{\cdots }+\frac{1}{2n-1}\right),$$ $n=1,2,\mathrm{\dots }$.

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Symbols:

$\gamma$: Euler’s constant, $\psi \left(z\right)$: psi (or digamma) function, $\ln z$: principal branch of logarithm function and $n$: nonnegative integer

A&S Ref:

6.3.4

Permalink:

http://dlmf.nist.gov/5.4.E15

Encodings:

pMML, png, TeX

See also:

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

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5.4.16 $$\mathrm{\Im }\psi \left(\mathrm{i}y\right)=\frac{1}{2y}+\frac{\pi }{2}\coth \left(\pi y\right),$$

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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, $\coth z$: hyperbolic cotangent function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit and $y$: real variable

A&S Ref:

6.3.11

Permalink:

http://dlmf.nist.gov/5.4.E16

Encodings:

pMML, png, TeX

See also:

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

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5.4.17 $$\mathrm{\Im }\psi \left(\frac{1}{2}+\mathrm{i}y\right)=\frac{\pi }{2}\tanh \left(\pi y\right),$$

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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, $\tanh z$: hyperbolic tangent function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit and $y$: real variable

A&S Ref:

6.3.12

Permalink:

http://dlmf.nist.gov/5.4.E17

Encodings:

pMML, png, TeX

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:

pMML, png, TeX

See also:

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

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6: 25.1 Special Notation

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$k,m,n$	nonnegative integers.
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$\psi \left(x\right)$	digamma function ${\mathrm{\Gamma }}^{\prime }\left(x\right)/\mathrm{\Gamma }\left(x\right)$ except in §25.16. See §5.2(i).
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7: 5.5 Functional Relations

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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:

pMML, png, TeX

See also:

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

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5.5.4 $$\psi \left(z\right)-\psi \left(1-z\right)=-\pi /\tan \left(\pi z\right),$$ $z\ne 0,\pm 1,\mathrm{\dots }$.

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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, $\tan z$: tangent function and $z$: complex variable

A&S Ref:

6.3.7 (without the condition on $z$.)

Permalink:

http://dlmf.nist.gov/5.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §5.5(ii), §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:

pMML, png, TeX

See also:

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

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5.5.9 $$\psi \left(nz\right)=\frac{1}{n}\sum _{k=0}^{n-1}\psi \left(z+\frac{k}{n}\right)+\ln n.$$

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Symbols:

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

Referenced by:

§5.5(iii)

Permalink:

http://dlmf.nist.gov/5.5.E9

Encodings:

pMML, png, TeX

See also:

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

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8: 5.3 Graphics

§5.3 Graphics

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►►

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

… ►In the graphics shown in this subsection, both the height and color correspond to the absolute value of the function. … ►

►

►

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

9: 14.11 Derivatives with Respect to Degree or Order

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14.11.3 $${𝖠}_{\nu }^{\mu }(x)=\sin \left(\nu \pi \right){\left(\frac{1+x}{1-x}\right)}^{\mu /2}\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-\frac{1}{2}x\right)}^k\mathrm{\Gamma }\left(k-\nu \right)\mathrm{\Gamma }\left(k+\nu +1\right)}{k!\mathrm{\Gamma }\left(k-\mu +1\right)}\left(\psi \left(k+\nu +1\right)-\psi \left(k-\nu \right)\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\sin z$: sine function, $x$: real variable, $\mu$: general order, $\nu$: general degree and ${𝖠}_{\nu }^{\mu }(x)$

Referenced by:

§14.11

Permalink:

http://dlmf.nist.gov/14.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.11 and Ch.14

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14.11.4 $${\frac{\partial }{\partial \mu }{𝖯}_{\nu }^{\mu }\left(x\right)|}_{\mu =0}=\left(\psi \left(-\nu \right)-\pi \cot \left(\nu \pi \right)\right){𝖯}_{\nu }\left(x\right)+{𝖰}_{\nu }\left(x\right),$$

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Symbols:

: Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\psi \left(z\right)$: psi (or digamma) function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.11, §14.11

Permalink:

http://dlmf.nist.gov/14.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.11 and Ch.14

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14.11.5 $${\frac{\partial }{\partial \mu }{𝖰}_{\nu }^{\mu }\left(x\right)|}_{\mu =0}=-\frac{1}{4}{\pi }^2{𝖯}_{\nu }\left(x\right)+\left(\psi \left(-\nu \right)-\pi \cot \left(\nu \pi \right)\right){𝖰}_{\nu }\left(x\right).$$

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Symbols:

: Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\psi \left(z\right)$: psi (or digamma) function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.11

Permalink:

http://dlmf.nist.gov/14.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §14.11 and Ch.14

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10: 5.2 Definitions

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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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