psi function

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31: Bibliography

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Numerical computation of Tricomi’s psi function by the trapezoidal rule. Computing 39 (3), pp. 271–279.

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External Links:

ISSN 0010-485X, Document, MathReview, ZentralBlatt

Referenced by:

§13.29(iii).

Encodings:

BibTeX

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Algorithm 610. A portable FORTRAN subroutine for derivatives of the psi function. ACM Trans. Math. Software 9 (4), pp. 494–502.

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Notes:

Single and double precision, maximum accuracy 18S.

External Links:

ISSN 0098-3500, Document, GAMS (module 610; package TOMS), MathReview Entry, ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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32: 18.39 Applications in the Physical Sciences

… ► $\mathscr{H}$ also controls time evolution of the wave function $\mathrm{\Psi }(x,t)$ via the time-dependent Schrödinger equation, … ►

$c_n=⟨\chi ,{\psi }_n⟩,$

… ►The finite system of functions ${\psi }_n$ is orthonormal in $L^2(ℝ,dx)$, see (18.34.7_3). … ►An alternative, and often used, form of (18.39.25) is that for the spherical radial function ${\psi }_{n,l}(r)=r{R}_{n,l}(r)$, … ►The functions ${\psi }_{p,l}(r)$ satisfy the equation, …

33: 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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34: 10.15 Derivatives with Respect to Order

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

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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

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35: 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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36: 2.8 Differential Equations with a Parameter

… ►where … ►In Cases I and II the asymptotic solutions are in terms of the functions that satisfy (2.8.8) with $\psi (\xi )=0$. … ►in which $\xi$ ranges over a bounded or unbounded interval or domain $𝚫$, and $\psi (\xi )$ is $C^{\mathrm{\infty }}$ or analytic on $𝚫$. … ►Again, $u>0$ and $\psi (\xi )$ is $C^{\mathrm{\infty }}$ on $({\alpha }_1,{\alpha }_2)$. … ►Also, $\psi (\xi )$ is $C^{\mathrm{\infty }}$ on $({\alpha }_1,{\alpha }_2)$, and $u>0$. …

37: Bibliography W

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The zeros of Euler’s psi function and its derivatives. J. Math. Anal. Appl. 332 (1), pp. 607–616.

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External Links:

ISSN 0022-247X, Document, MathReview, ZentralBlatt (Mehdi Hassani)

Referenced by:

§5.4(iii).

Encodings:

BibTeX

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38: 36.12 Uniform Approximation of Integrals

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36.12.3 $$I(𝐲,k)=\frac{\exp \left(\mathrm{i}kA(𝐲)\right)}{k^{1/(K+2)}}\sum _{m=0}^K\frac{a_m(𝐲)}{k^{m/(K+2)}}\left({\delta }_{m,0}-\left(1-{\delta }_{m,0}\right)\mathrm{i}\frac{\partial }{\partial z_m}\right){\mathrm{\Psi }}_K\left(𝐳(𝐲;k)\right)\left(1+O\left(\frac{1}{k}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, ${\delta }_{j,k}$: Kronecker delta, ${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: integer, $k$: variable, $K$: codimension, $I(𝐲,k)$: oscillatory integral, $a_m(𝐲)$: function, $A(𝐲)$: function, $𝐳(𝐲,k)$: function and $z_m(𝐲,k)$: function

Permalink:

http://dlmf.nist.gov/36.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §36.12(i), §36.12 and Ch.36

… ►For example, the diffraction catastrophe ${\mathrm{\Psi }}_2(x,y;k)$ defined by (36.2.10), and corresponding to the Pearcey integral (36.2.14), can be approximated by the Airy function ${\mathrm{\Psi }}_1\left(\xi (x,y;k)\right)$ when $k$ is large, provided that $x$ and $y$ are not small. …

39: 9.18 Tables

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Miller (1946) tabulates $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$ for $x=-20(.01)2;$ ${\log }_{10}\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)/\mathrm{Ai}\left(x\right)$ for $x=0(.1)25(1)75$; $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$ for $x=-10(.1)2.5$; ${\log }_{10}\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)/\mathrm{Bi}\left(x\right)$ for $x=0(.1)10$; $M\left(x\right)$, $N\left(x\right)$, $\theta \left(x\right)$, $\varphi \left(x\right)$ (respectively $F(x)$, $G(x)$, $\chi (x)$, $\psi (x)$) for $x=-80(1)-30(.1)0$. Precision is generally 8D; slightly less for some of the auxiliary functions. Extracts from these tables are included in Abramowitz and Stegun (1964, Chapter 10), together with some auxiliary functions for large arguments.

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40: 17.4 Basic Hypergeometric Functions

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§17.4(ii) ${}_r{}^{}\psi _s^{}$ Functions

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17.4.3 $${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,a_2,\mathrm{\dots },a_r}{b_1,b_2,\mathrm{\dots },b_s};q,z)={}_r{}^{}\psi _s^{}(a_1,a_2,\mathrm{\dots },a_r;b_1,b_2,\mathrm{\dots },b_s;q,z)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{(a_1,a_2,\mathrm{\dots },a_r;q)}_n{(-1)}^{(s-r)n}{q}^{(s-r)\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}{z}^n}{{(b_1,b_2,\mathrm{\dots },b_s;q)}_n}=\sum _{n=0}^{\mathrm{\infty }}\frac{{(a_1,a_2,\mathrm{\dots },a_r;q)}_n{(-1)}^{(s-r)n}{q}^{(s-r)\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}{z}^n}{{(b_1,b_2,\mathrm{\dots },b_s;q)}_n}+\sum _{n=1}^{\mathrm{\infty }}\frac{{(q/b_1,q/b_2,\mathrm{\dots },q/b_s;q)}_n}{{(q/a_1,q/a_2,\mathrm{\dots },q/a_r;q)}_n}{\left(\frac{b_1{b}_2\mathrm{\cdots }{b}_s}{a_1{a}_2\mathrm{\cdots }{a}_{r}z}\right)}^n.$$

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Defines:

${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $n$: nonnegative integer, $r$: nonnegative integer, $s$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/17.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §17.4(ii), §17.4 and Ch.17

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17.4.4 $$\underset{q\to 1-}{lim}{}_r{}^{}\psi _r^{}(\genfrac{}{}{0pt}{}{q^{a_1},q^{a_2},\mathrm{\dots },q^{a_r}}{q^{b_1},q^{b_2},\mathrm{\dots },q^{b_r}};q,z)={}_r{}^{}H_r^{}(\genfrac{}{}{0pt}{}{a_1,a_2,\mathrm{\dots },a_r}{b_1,b_2,\mathrm{\dots },b_r};z).$$

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Symbols:

${}_p{}^{}H_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: bilateral hypergeometric function, ${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function, $q$: complex base, $r$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/17.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §17.4(ii), §17.4 and Ch.17

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