digamma function

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11: 14.17 Integrals

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14.17.10 $${\int }_{-1}^1{𝖯}_{\nu }\left(x\right){𝖯}_{\lambda }\left(x\right)dx=\frac{2\left(2\sin \left(\nu \pi \right)\sin \left(\lambda \pi \right)\left(\psi \left(\nu +1\right)-\psi \left(\lambda +1\right)\right)+\pi \sin \left((\lambda -\nu )\pi \right)\right)}{{\pi }^2(\lambda -\nu )(\lambda +\nu +1)},$$ $\lambda \ne \nu$ or $-\nu -1$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\psi \left(z\right)$: psi (or digamma) function, $\int$: integral, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $\sin z$: sine function, $x$: real variable and $\nu$: general degree

A&S Ref:

8.14.4

Permalink:

http://dlmf.nist.gov/14.17.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §14.17(iv), §14.17 and Ch.14

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14.17.13 $${\int }_{-1}^1{\left({𝖰}_{\nu }\left(x\right)\right)}^{2}dx=\frac{{\pi }^2-2\left(1+{\cos }^2\left(\nu \pi \right)\right){\psi }^{\prime }\left(\nu +1\right)}{2(2\nu +1)},$$ $\nu \ne -\frac{1}{2}$ or $-1,-2,-3,\mathrm{\dots }$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\psi \left(z\right)$: psi (or digamma) function, $\int$: integral, ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $x$: real variable and $\nu$: general degree

A&S Ref:

8.14.7

Permalink:

http://dlmf.nist.gov/14.17.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §14.17(iv), §14.17 and Ch.14

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14.17.15 $${\int }_{-1}^1{𝖯}_{\nu }\left(x\right){𝖰}_{\nu }\left(x\right)dx=-\frac{\sin \left(2\nu \pi \right){\psi }^{\prime }\left(\nu +1\right)}{\pi (2\nu +1)},$$ $\mathrm{\Re }\nu >0$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\psi \left(z\right)$: psi (or digamma) function, $\int$: integral, $\mathrm{\Re }$: real part, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $\sin z$: sine function, $x$: real variable and $\nu$: general degree

A&S Ref:

8.14.9

Permalink:

http://dlmf.nist.gov/14.17.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §14.17(iv), §14.17 and Ch.14

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14.17.19 $${\int }_1^{\mathrm{\infty }}{Q}_{\nu }\left(x\right)Q_{\lambda }\left(x\right)dx=\frac{\psi \left(\lambda +1\right)-\psi \left(\nu +1\right)}{(\lambda -\nu )(\lambda +\nu +1)},$$ $\mathrm{\Re }\left(\lambda +\nu \right)>-1$, $\lambda \ne \nu$, $\lambda$ and $\nu \ne -1,-2,-3,\mathrm{\dots }$.

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Symbols:

$dx$: differential of $x$ , $\psi \left(z\right)$: psi (or digamma) function, $\int$: integral, $\mathrm{\Re }$: real part, $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: Legendre function of the second kind, $x$: real variable and $\nu$: general degree

A&S Ref:

8.14.2

Permalink:

http://dlmf.nist.gov/14.17.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §14.17(iv), §14.17 and Ch.14

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14.17.20 $${\int }_1^{\mathrm{\infty }}{(Q_{\nu }\left(x\right))}^{2}dx=\frac{{\psi }^{\prime }\left(\nu +1\right)}{2\nu +1},$$ $\mathrm{\Re }\nu >-\frac{1}{2}$.

ⓘ

Symbols:

$dx$: differential of $x$ , $\psi \left(z\right)$: psi (or digamma) function, $\int$: integral, $\mathrm{\Re }$: real part, $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: Legendre function of the second kind, $x$: real variable and $\nu$: general degree

A&S Ref:

8.14.3

Permalink:

http://dlmf.nist.gov/14.17.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §14.17(iv), §14.17 and Ch.14

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12: 33.19 Power-Series Expansions in $r$

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33.19.3 $$2\pi h(ϵ,\mathrm{\ell };r)=\sum _{k=0}^{2\mathrm{\ell }}\frac{(2\mathrm{\ell }-k)!{\gamma }_k}{k!}{(2r)}^{k-\mathrm{\ell }}-\sum _{k=0}^{\mathrm{\infty }}{\delta }_k{r}^{k+\mathrm{\ell }+1}-A(ϵ,\mathrm{\ell })\left(2\ln |2r/\kappa |+\mathrm{\Re }\psi \left(\mathrm{\ell }+1+\kappa \right)+\mathrm{\Re }\psi \left(-\mathrm{\ell }+\kappa \right)\right)f(ϵ,\mathrm{\ell };r),$$ $r\ne 0$.

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Symbols:

$f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $h(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter, $\kappa$: quantity, $A(ϵ,\mathrm{\ell })$: function, ${\gamma }_k$: term and ${\delta }_k$: term

Referenced by:

§33.19

Permalink:

http://dlmf.nist.gov/33.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §33.19 and Ch.33

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13: 5.6 Inequalities

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5.6.5 $$\exp \left((1-s)\psi \left(x+s^{1/2}\right)\right)\le \frac{\mathrm{\Gamma }\left(x+1\right)}{\mathrm{\Gamma }\left(x+s\right)}\le \exp \left((1-s)\psi \left(x+\frac{1}{2}(s+1)\right)\right),$$ .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\psi \left(z\right)$: psi (or digamma) function, $\exp z$: exponential function, $s$: real or complex variable and $x$: real variable

Referenced by:

§5.6(i)

Permalink:

http://dlmf.nist.gov/5.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §5.6(i), §5.6(i), §5.6 and Ch.5

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14: 25.8 Sums

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25.8.5 $$\sum _{k=2}^{\mathrm{\infty }}\zeta \left(k\right)z^k=-\gamma z-z\psi \left(1-z\right),$$ .

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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

Keywords:

infinite series

Source:

Erdélyi et al. (1953a, (1.17.5), p. 45); with $z\mapsto -z$

Referenced by:

(25.8.7)

Permalink:

http://dlmf.nist.gov/25.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §25.8 and Ch.25

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15: 6.6 Power Series

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6.6.3 $$E_1\left(z\right)=-\ln z+{\mathrm{e}}^{-z}\sum _{n=0}^{\mathrm{\infty }}\frac{z^n}{n!}\psi \left(n+1\right),$$

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Symbols:

$\psi \left(z\right)$: psi (or digamma) function, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§2.5(iii), §6.6, §8.19(iv)

Permalink:

http://dlmf.nist.gov/6.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.6 and Ch.6

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

16: 5.19 Mathematical Applications

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5.19.3 $$S=\psi \left(\frac{1}{2}\right)-2\psi \left(\frac{2}{3}\right)-\gamma =3\ln 3-2\ln 2-\frac{1}{3}\pi \sqrt{3}.$$

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Symbols:

$\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function and $\ln z$: principal branch of logarithm function

Permalink:

http://dlmf.nist.gov/5.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.19(i), §5.19(i), §5.19 and Ch.5

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17: 10.31 Power Series

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10.31.1 $$K_n\left(z\right)=\frac{1}{2}{(\frac{1}{2}z)}^{-n}\sum _{k=0}^{n-1}\frac{(n-k-1)!}{k!}{(-\frac{1}{4}{z}^2)}^k+{(-1)}^{n+1}\ln \left(\frac{1}{2}z\right)I_n\left(z\right)+{(-1)}^n\frac{1}{2}{(\frac{1}{2}z)}^n\sum _{k=0}^{\mathrm{\infty }}\left(\psi \left(k+1\right)+\psi \left(n+k+1\right)\right)\frac{{(\frac{1}{4}{z}^2)}^k}{k!(n+k)!},$$

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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:

9.6.11

Referenced by:

§10.30(i), §10.65(ii)

Permalink:

http://dlmf.nist.gov/10.31.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.31 and Ch.10

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18: 10.65 Power Series

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10.65.3 $${\ker }_{n}x=\frac{1}{2}{(\frac{1}{2}x)}^{-n}\sum _{k=0}^{n-1}\frac{(n-k-1)!}{k!}\cos \left(\frac{3}{4}n\pi +\frac{1}{2}k\pi \right){(\frac{1}{4}{x}^2)}^k-\ln \left(\frac{1}{2}x\right){\mathrm{ber}}_{n}x+\frac{1}{4}\pi {\mathrm{bei}}_{n}x+\frac{1}{2}{(\frac{1}{2}x)}^n\sum _{k=0}^{\mathrm{\infty }}\frac{\psi \left(k+1\right)+\psi \left(n+k+1\right)}{k!(n+k)!}\cos \left(\frac{3}{4}n\pi +\frac{1}{2}k\pi \right){(\frac{1}{4}{x}^2)}^k,$$

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Symbols:

${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin 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, $n$: integer, $k$: nonnegative integer and $x$: real variable

A&S Ref:

9.9.11

Permalink:

http://dlmf.nist.gov/10.65.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.65(ii), §10.65 and Ch.10

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10.65.4 $${\mathrm{kei}}_{n}x=-\frac{1}{2}{(\frac{1}{2}x)}^{-n}\sum _{k=0}^{n-1}\frac{(n-k-1)!}{k!}\sin \left(\frac{3}{4}n\pi +\frac{1}{2}k\pi \right){(\frac{1}{4}{x}^2)}^k-\ln \left(\frac{1}{2}x\right){\mathrm{bei}}_{n}x-\frac{1}{4}\pi {\mathrm{ber}}_{n}x+\frac{1}{2}{(\frac{1}{2}x)}^n\sum _{k=0}^{\mathrm{\infty }}\frac{\psi \left(k+1\right)+\psi \left(n+k+1\right)}{k!(n+k)!}\sin \left(\frac{3}{4}n\pi +\frac{1}{2}k\pi \right){(\frac{1}{4}{x}^2)}^k.$$

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Symbols:

${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $\sin z$: sine function, $n$: integer, $k$: nonnegative integer and $x$: real variable

A&S Ref:

9.9.11

Permalink:

http://dlmf.nist.gov/10.65.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.65(ii), §10.65 and Ch.10

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19: 5.9 Integral Representations

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5.9.12 $$\psi \left(z\right)={\int }_0^{\mathrm{\infty }}\left(\frac{{\mathrm{e}}^{-t}}{t}-\frac{{\mathrm{e}}^{-zt}}{1-{\mathrm{e}}^{-t}}\right)dt,$$

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Symbols:

$dx$: differential of $x$ , $\psi \left(z\right)$: psi (or digamma) function, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $z$: complex variable

A&S Ref:

6.3.21

Permalink:

http://dlmf.nist.gov/5.9.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §5.9(ii), §5.9 and Ch.5

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5.9.13 $$\psi \left(z\right)=\ln z+{\int }_0^{\mathrm{\infty }}\left(\frac{1}{t}-\frac{1}{1-{\mathrm{e}}^{-t}}\right){\mathrm{e}}^{-tz}dt,$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\psi \left(z\right)$: psi (or digamma) function, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\ln z$: principal branch of logarithm function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/5.9.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §5.9(ii), §5.9 and Ch.5

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5.9.14 $$\psi \left(z\right)={\int }_0^{\mathrm{\infty }}\left({\mathrm{e}}^{-t}-\frac{1}{{(1+t)}^z}\right)\frac{dt}{t},$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\psi \left(z\right)$: psi (or digamma) function, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $z$: complex variable

A&S Ref:

6.3.21

Permalink:

http://dlmf.nist.gov/5.9.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §5.9(ii), §5.9 and Ch.5

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5.9.15 $$\psi \left(z\right)=\ln z-\frac{1}{2z}-2{\int }_0^{\mathrm{\infty }}\frac{tdt}{(t^2+z^2)({\mathrm{e}}^{2\pi t}-1)}.$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\psi \left(z\right)$: psi (or digamma) function, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

6.3.21

Referenced by:

§5.9(ii)

Permalink:

http://dlmf.nist.gov/5.9.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §5.9(ii), §5.9 and Ch.5

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5.9.17 $$\psi \left(z+1\right)=-\gamma +\frac{1}{2\pi \mathrm{i}}{\int }_{-c-\mathrm{\infty }\mathrm{i}}^{-c+\mathrm{\infty }\mathrm{i}}\frac{\pi z^{-s-1}}{\sin \left(\pi s\right)}\zeta \left(-s\right)ds,$$

ⓘ

Symbols:

$\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\psi \left(z\right)$: psi (or digamma) function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $s$: real or complex variable and $z$: complex variable

Referenced by:

§5.9(ii)

Permalink:

http://dlmf.nist.gov/5.9.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §5.9(ii), §5.9 and Ch.5

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20: 25.5 Integral Representations

… ►In (25.5.15)–(25.5.19), , $\psi \left(x\right)$ is the digamma function, and $\gamma$ is Euler’s constant (§5.2). … ►

25.5.15 $$\zeta \left(s\right)=\frac{1}{s-1}+\frac{\sin \left(\pi s\right)}{\pi }{\int }_0^{\mathrm{\infty }}(\ln \left(1+x\right)-\psi \left(1+x\right))x^{-s}dx,$$

ⓘ

Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\psi \left(z\right)$: psi (or digamma) function, $\int$: integral, $\ln z$: principal branch of logarithm function, $\sin z$: sine function, $x$: real variable and $s$: complex variable

Keywords:

improper integral, integral representation

Source:

de Bruijn (1937, (18), p. 175)

Referenced by:

§25.5(ii)

Permalink:

http://dlmf.nist.gov/25.5.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §25.5(ii), §25.5 and Ch.25

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25.5.17 $$\zeta \left(1+s\right)=\frac{\sin \left(\pi s\right)}{\pi }{\int }_0^{\mathrm{\infty }}\left(\gamma +\psi \left(1+x\right)\right)x^{-s-1}dx,$$

ⓘ

Symbols:

$\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\psi \left(z\right)$: psi (or digamma) function, $\int$: integral, $\sin z$: sine function, $x$: real variable and $s$: complex variable

Keywords:

improper integral, integral representation

Source:

de Bruijn (1937, (22), p. 177)

Permalink:

http://dlmf.nist.gov/25.5.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §25.5(ii), §25.5 and Ch.25

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25.5.18 $$\zeta \left(1+s\right)=\frac{\sin \left(\pi s\right)}{\pi s}{\int }_0^{\mathrm{\infty }}{\psi }^{\prime }\left(1+x\right)x^{-s}dx,$$

ⓘ

Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\psi \left(z\right)$: psi (or digamma) function, $\int$: integral, $\sin z$: sine function, $x$: real variable and $s$: complex variable

Keywords:

improper integral, integral representation

Source:

de Bruijn (1937, p. 177)

Permalink:

http://dlmf.nist.gov/25.5.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §25.5(ii), §25.5 and Ch.25

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25.5.19 $$\zeta \left(m+s\right)={(-1)}^{m-1}\frac{\mathrm{\Gamma }\left(s\right)\sin \left(\pi s\right)}{\pi \mathrm{\Gamma }\left(m+s\right)}{\int }_0^{\mathrm{\infty }}{\psi }^{(m)}\left(1+x\right)x^{-s}dx,$$ $m=1,2,3,\mathrm{\dots }$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\psi \left(z\right)$: psi (or digamma) function, $\int$: integral, $\sin z$: sine function, $m$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

improper integral, integral representation

Source:

de Bruijn (1937, p. 178)

Referenced by:

§25.5(ii)

Permalink:

http://dlmf.nist.gov/25.5.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §25.5(ii), §25.5 and Ch.25

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