digamma function

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21: 19.12 Asymptotic Approximations

… ►With $\psi \left(x\right)$ denoting the digamma function (§5.2(i)) in this subsection, the asymptotic behavior of $K\left(k\right)$ and $E\left(k\right)$ near the singularity at $k=1$ is given by the following convergent series: …

22: 10.38 Derivatives with Respect to Order

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10.38.1 $$\frac{\partial I_{\pm \nu }\left(z\right)}{\partial \nu }=\pm I_{\pm \nu }\left(z\right)\ln \left(\frac{1}{2}z\right)\mp {(\frac{1}{2}z)}^{\pm \nu }\sum _{k=0}^{\mathrm{\infty }}\frac{\psi \left(k+1\pm \nu \right)}{\mathrm{\Gamma }\left(k+1\pm \nu \right)}\frac{{(\frac{1}{4}{z}^2)}^k}{k!},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\ln z$: principal branch of logarithm function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.42

Referenced by:

§10.38, §10.38, Erratum (V1.0.17) for Equations (10.15.1), (10.38.1)

Permalink:

http://dlmf.nist.gov/10.38.E1

Encodings:

pMML, png, TeX

Addition (effective with 1.0.17):

This equation has been generalized to include the additional case of $\partial I_{-\nu }\left(z\right)/\partial \nu$ because it will help the user who wants to combine it with (10.38.2).

See also:

Annotations for §10.38 and Ch.10

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23: 10.8 Power Series

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10.8.1 $$Y_n\left(z\right)=-\frac{{(\frac{1}{2}z)}^{-n}}{\pi }\sum _{k=0}^{n-1}\frac{(n-k-1)!}{k!}{\left(\frac{1}{4}{z}^2\right)}^k+\frac{2}{\pi }\ln \left(\frac{1}{2}z\right)J_n\left(z\right)-\frac{{(\frac{1}{2}z)}^n}{\pi }\sum _{k=0}^{\mathrm{\infty }}(\psi \left(k+1\right)+\psi \left(n+k+1\right))\frac{{(-\frac{1}{4}{z}^2)}^k}{k!(n+k)!},$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $n$: integer, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

9.1.11

Referenced by:

§10.19(i), §10.7(i)

Permalink:

http://dlmf.nist.gov/10.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.8 and Ch.10

…

24: 13.2 Definitions and Basic Properties

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13.2.19 $$U(a,1,z)=-\frac{1}{\mathrm{\Gamma }\left(a\right)}\left(\ln z+\psi \left(a\right)+2\gamma \right)+O\left(z\ln z\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\gamma$: Euler’s constant, $U(a,b,z)$: Kummer confluent hypergeometric function, $\psi \left(z\right)$: psi (or digamma) function, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

13.5.9 (as corrected in later editions)

Referenced by:

§13.2(iii)

Permalink:

http://dlmf.nist.gov/13.2.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §13.2(iii), §13.2 and Ch.13

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13.2.27 $$\sum _{k=1}^n\frac{n!(k-1)!}{(n-k)!{\left(1-a\right)}_k}{z}^{-k}-\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_k}{{\left(n+1\right)}_{k}k!}{z}^k\left(\ln z+\psi \left(a+k\right)-\psi \left(1+k\right)-\psi \left(n+k+1\right)\right),$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $n$: nonnegative integer and $z$: complex variable

Referenced by:

§13.2(v)

Permalink:

http://dlmf.nist.gov/13.2.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §13.2(v), §13.2 and Ch.13

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13.2.28 $$\sum _{k=1}^n\frac{n!(k-1)!}{(n-k)!{\left(1-a\right)}_k}{z}^{-k}-\sum _{k=0}^{-a}\frac{{\left(a\right)}_k}{{\left(n+1\right)}_{k}k!}{z}^k\left(\ln z+\psi \left(1-a-k\right)-\psi \left(1+k\right)-\psi \left(n+k+1\right)\right)+{(-1)}^{1-a}(-a)!\sum _{k=1-a}^{\mathrm{\infty }}\frac{(k-1+a)!}{{\left(n+1\right)}_{k}k!}{z}^k,$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.2.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §13.2(v), §13.2 and Ch.13

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13.2.30 $$\sum _{k=1}^{n+1}\frac{(n+1)!(k-1)!}{(n-k+1)!{\left(-a-n\right)}_k}{z}^{n-k+1}-\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a+n+1\right)}_k}{{\left(n+2\right)}_{k}k!}{z}^{n+k+1}\left(\ln z+\psi \left(a+n+k+1\right)-\psi \left(1+k\right)-\psi \left(n+k+2\right)\right),$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.2.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §13.2(v), §13.2 and Ch.13

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13.2.31 $$\sum _{k=1}^{n+1}\frac{(n+1)!(k-1)!}{(n-k+1)!{\left(-a-n\right)}_k}{z}^{n-k+1}-\sum _{k=0}^{-a-n-1}\frac{{\left(a+n+1\right)}_k}{{\left(n+2\right)}_{k}k!}{z}^{n+k+1}\left(\ln z+\psi \left(-a-n-k\right)-\psi \left(1+k\right)-\psi \left(n+k+2\right)\right)+{(-1)}^{n-a}(-a-n-1)!\sum _{k=-a-n}^{\mathrm{\infty }}\frac{(k+a+n)!}{{\left(n+2\right)}_{k}k!}{z}^{n+k+1},$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.2.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §13.2(v), §13.2 and Ch.13

…

25: 6.10 Other Series Expansions

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6.10.7 $$a_n=(2n+1)\left(1-{(-1)}^n+\psi \left(n+1\right)-\psi \left(1\right)\right),$$

ⓘ

Defines:

$a_n$: coefficients (locally)

Symbols:

$\psi \left(z\right)$: psi (or digamma) function and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/6.10.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §6.10(ii), §6.10 and Ch.6

►and $\psi$ denotes the logarithmic derivative of the gamma function (§5.2(i)). …

26: 33.6 Power-Series Expansions in $\rho$

… ►

33.6.5 $$H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )=\frac{{\mathrm{e}}^{\pm \mathrm{i}{\theta }_{\mathrm{\ell }}(\eta ,\rho )}}{(2\mathrm{\ell }+1)!\mathrm{\Gamma }\left(-\mathrm{\ell }\pm \mathrm{i}\eta \right)}\left(\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_k}{{\left(2\mathrm{\ell }+2\right)}_{k}k!}{(\mp 2\mathrm{i}\rho )}^{a+k}\left(\ln \left(\mp 2\mathrm{i}\rho \right)+\psi \left(a+k\right)-\psi \left(1+k\right)-\psi \left(2\mathrm{\ell }+2+k\right)\right)-\sum _{k=1}^{2\mathrm{\ell }+1}\frac{(2\mathrm{\ell }+1)!(k-1)!}{(2\mathrm{\ell }+1-k)!{\left(1-a\right)}_k}{(\mp 2\mathrm{i}\rho )}^{a-k}\right),$$

ⓘ

Symbols:

${\theta }_{\mathrm{\ell }}(\eta ,\rho )$: phase of Coulomb functions, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\psi \left(z\right)$: psi (or digamma) function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter and $a$: variable

A&S Ref:

14.1.14 (equivalent)

Referenced by:

§33.13, §33.6, §33.6, Erratum (V1.0.19) for Equation (33.6.5)

Permalink:

http://dlmf.nist.gov/33.6.E5

Encodings:

pMML, png, TeX

Errata (effective with 1.0.19):

On the right-hand side the factor in the denominator was incorrectly written as $\mathrm{\Gamma }\left(-\mathrm{\ell }+\mathrm{i}\eta \right)$. This has been corrected to $\mathrm{\Gamma }\left(-\mathrm{\ell }\pm \mathrm{i}\eta \right)$.

Reported 2018-05-15 by Ian Thompson

See also:

Annotations for §33.6 and Ch.33

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27: 10.44 Sums

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10.44.6 $$K_n\left(z\right)=\frac{n!{(\frac{1}{2}z)}^{-n}}{2}\sum _{k=0}^{n-1}{(-1)}^k\frac{{(\frac{1}{2}z)}^k{I}_k\left(z\right)}{k!(n-k)}+{(-1)}^{n-1}\left(\ln \left(\frac{1}{2}z\right)-\psi \left(n+1\right)\right)I_n\left(z\right)+{(-1)}^n\sum _{k=1}^{\mathrm{\infty }}\frac{(n+2k)I_{n+2k}\left(z\right)}{k(n+k)},$$

ⓘ

Symbols:

$\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\ln z$: principal branch of logarithm function, $n$: integer, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

9.6.53

Permalink:

http://dlmf.nist.gov/10.44.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.44(iii), §10.44 and Ch.10

…

28: 10.15 Derivatives with Respect to Order

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10.15.1 $$\frac{\partial J_{\pm \nu }\left(z\right)}{\partial \nu }=\pm J_{\pm \nu }\left(z\right)\ln \left(\frac{1}{2}z\right)\mp {(\frac{1}{2}z)}^{\pm \nu }\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{\psi \left(k+1\pm \nu \right)}{\mathrm{\Gamma }\left(k+1\pm \nu \right)}\frac{{(\frac{1}{4}{z}^2)}^k}{k!},$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.64

Referenced by:

§10.15, §10.38, Erratum (V1.0.17) for Equations (10.15.1), (10.38.1)

Permalink:

http://dlmf.nist.gov/10.15.E1

Encodings:

pMML, png, TeX

Addition (effective with 1.0.17):

This equation has been generalized to include the additional case of $\partial J_{-\nu }\left(z\right)/\partial \nu$ because it will help the user who wants to combine it with (10.15.2).

See also:

Annotations for §10.15, §10.15 and Ch.10

…

29: 25.11 Hurwitz Zeta Function

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25.11.12 $$\zeta (n+1,a)=\frac{{(-1)}^{n+1}{\psi }^{(n)}\left(a\right)}{n!},$$ $n=1,2,3,\mathrm{\dots }$.

ⓘ

Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $n$: nonnegative integer and $a$: real or complex parameter

Keywords:

special value

Source:

Erdélyi et al. (1953a, (1.11.8), p. 29)

Permalink:

http://dlmf.nist.gov/25.11.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(v), §25.11 and Ch.25

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25.11.21 $${\zeta }^{\prime }(1-2n,\frac{h}{k})=\frac{(\psi \left(2n\right)-\ln \left(2\pi k\right))B_{2n}\left(h/k\right)}{2n}-\frac{(\psi \left(2n\right)-\ln \left(2\pi \right))B_{2n}}{2n{k}^{2n}}+\frac{{(-1)}^{n+1}\pi }{{(2\pi k)}^{2n}}\sum _{r=1}^{k-1}\sin \left(\frac{2\pi rh}{k}\right){\psi }^{(2n-1)}\left(\frac{r}{k}\right)+\frac{{(-1)}^{n+1}2\cdot (2n-1)!}{{(2\pi k)}^{2n}}\sum _{r=1}^{k-1}\cos \left(\frac{2\pi rh}{k}\right){\zeta }^{\prime }(2n,\frac{r}{k})+\frac{{\zeta }^{\prime }\left(1-2n\right)}{k^{2n}},$$

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $B_n\left(x\right)$: Bernoulli polynomials, $\zeta (s,a)$: Hurwitz zeta function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $\sin z$: sine function, $n$: nonnegative integer, $h$: integer and $k$: integer

Keywords:

derivative

Source:

Miller and Adamchik (1998, (5), p. 203)

Notes:

According to the source, (25.11.21) is valid only for . Their derivation, however, is based on (25.11.16) which is true when $h=k$ as well. Alternatively, one can substitute $h=k$ into (25.11.21) and simplify via (25.11.24), (25.6.2) and (25.6.15) confirming that (25.11.21) holds also in the case that $h=k$.

Referenced by:

(25.11.21)

Permalink:

http://dlmf.nist.gov/25.11.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(vi), §25.11(vi), §25.11 and Ch.25

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25.11.23 $${\zeta }^{\prime }(1-2n,\frac{1}{3})=-\frac{\pi (9^n-1)B_{2n}}{8n\sqrt{3}(3^{2n-1}-1)}-\frac{B_{2n}\ln 3}{4n\cdot 3^{2n-1}}-\frac{{(-1)}^n{\psi }^{(2n-1)}\left(\frac{1}{3}\right)}{2\sqrt{3}{(6\pi )}^{2n-1}}-\frac{\left(3^{2n-1}-1\right){\zeta }^{\prime }\left(1-2n\right)}{2\cdot 3^{2n-1}},$$ $n=1,2,3,\mathrm{\dots }$.

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\zeta (s,a)$: Hurwitz zeta function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $\ln z$: principal branch of logarithm function and $n$: nonnegative integer

Keywords:

derivative

Source:

Miller and Adamchik (1998, (18), p. 205)

Permalink:

http://dlmf.nist.gov/25.11.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(vi), §25.11(vi), §25.11 and Ch.25

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25.11.32 $${\int }_0^a{x}^n\psi \left(x\right)dx={(-1)}^{n-1}{\zeta }^{\prime }\left(-n\right)+{(-1)}^n{H}_n\frac{B_{n+1}}{n+1}-\sum _{k=0}^n{(-1)}^k\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)H_k\frac{B_{k+1}(a)}{k+1}{a}^{n-k}+\sum _{k=0}^n{(-1)}^k\left(\genfrac{}{}{0.0pt}{}{n}{k}\right){\zeta }^{\prime }(-k,a)a^{n-k},$$ $n=1,2,\mathrm{\dots }$, $\mathrm{\Re }a>0$,

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\zeta (s,a)$: Hurwitz zeta function, $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $dx$: differential of $x$ , $\psi \left(z\right)$: psi (or digamma) function, $H_n$: harmonic number, $\int$: integral, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable, $a$: real or complex parameter and $h(n)$: harmonic number

Keywords:

definite integral, integral representation

Source:

Adamchik (1998, (17), p. 197)

Referenced by:

Erratum (V1.1.4) for Notation

Permalink:

http://dlmf.nist.gov/25.11.E32

Encodings:

pMML, png, TeX

Notation (effective with 1.1.4):

The notation previously used for the harmonic number $h(n)$ (resp. $h(k)$) has been replaced to be $H_n$ (resp. $H_k$).

Suggested 2021-08-23 by Gergő Nemes

See also:

Annotations for §25.11(viii), §25.11 and Ch.25

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25.11.38 $$\sum _{k=1}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{n+k}{k}\right)\zeta (n+k+1,a)z^k=\frac{{(-1)}^n}{n!}\left({\psi }^{(n)}\left(a\right)-{\psi }^{(n)}\left(a-z\right)\right),$$ $n=1,2,3,\mathrm{\dots }$, $\mathrm{\Re }a>0$, .

ⓘ

Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer, $a$: real or complex parameter and $z$: complex variable

Keywords:

infinite series, series representation

Source:

Adamchik and Srivastava (1998, (2.29), p. 138)

Permalink:

http://dlmf.nist.gov/25.11.E38

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(xi), §25.11 and Ch.25

…

30: 8.19 Generalized Exponential Integral

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8.19.8 $$E_n\left(z\right)=\frac{{(-z)}^{n-1}}{(n-1)!}(\psi \left(n\right)-\ln z)-\sum _{\begin{array}{c}k=0\\ k\ne n-1\end{array}}^{\mathrm{\infty }}\frac{{(-z)}^k}{k!(1-n+k)},$$

ⓘ

Symbols:

$\psi \left(z\right)$: psi (or digamma) function, $E_p\left(z\right)$: generalized exponential integral, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $z$: complex variable, $k$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

5.1.12

Referenced by:

§8.19(iv)

Permalink:

http://dlmf.nist.gov/8.19.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(iv), §8.19 and Ch.8

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8.19.9 $$E_n\left(z\right)=\frac{{(-1)}^n{z}^{n-1}}{(n-1)!}\ln z+\frac{{\mathrm{e}}^{-z}}{(n-1)!}\sum _{k=1}^{n-1}{(-z)}^{k-1}\mathrm{\Gamma }\left(n-k\right)+\frac{{\mathrm{e}}^{-z}{(-z)}^{n-1}}{(n-1)!}\sum _{k=0}^{\mathrm{\infty }}\frac{z^k}{k!}\psi \left(k+1\right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\psi \left(z\right)$: psi (or digamma) function, $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $z$: complex variable, $k$: nonnegative integer and $n$: nonnegative integer

Referenced by:

§8.19(iv)

Permalink:

http://dlmf.nist.gov/8.19.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(iv), §8.19 and Ch.8

…