赞恩州立学院国际商务文凭证书〖办证V信ATV1819〗psiup

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1: 5.15 Polygamma Functions

… ►The functions ${\psi }^{(n)}\left(z\right)$, $n=1,2,\mathrm{\dots }$, are called the 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. … ►

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

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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

… ►

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

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Annotations for §5.15 and Ch.5

… ►For continued fractions for ${\psi }^{\prime }\left(z\right)$ and ${\psi }^{\prime \prime }\left(z\right)$ see Cuyt et al. (2008, pp. 231–238).

2: 5.16 Sums

… ►

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

►

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

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Annotations for §5.16 and Ch.5

…

3: 4.12 Generalized Logarithms and Exponentials

… ►Its inverse $\psi (x)$ is called a generalized logarithm. It, too, is strictly increasing when $0\le x\le 1$, and ►

4.12.3 $$\psi ({\mathrm{e}}^x)=1+\psi (x),$$ ,

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Symbols:

$\mathrm{e}$: base of natural logarithm, $x$: real variable and $\psi (x)$: generalized logarithm

Permalink:

http://dlmf.nist.gov/4.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.12 and Ch.4

►

4.12.4 $$\psi (0)=0.$$

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Symbols:

$\psi (x)$: generalized logarithm

Permalink:

http://dlmf.nist.gov/4.12.E4

Encodings:

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Annotations for §4.12 and Ch.4

… ►Both $\varphi (x)$ and $\psi (x)$ are continuously differentiable. …

4: 36.10 Differential Equations

… ►

§36.10(i) Equations for ${\mathrm{\Psi }}_K\left(𝐱\right)$

►In terms of the normal form (36.2.1) the ${\mathrm{\Psi }}_K\left(𝐱\right)$ satisfy the operator equation … ►

36.10.3 $$\frac{{\partial }^2{\mathrm{\Psi }}_1}{{\partial x}^2}-\frac{x}{3}{\mathrm{\Psi }}_1=0.$$

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Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §36.10(i), §36.10(i), §36.10 and Ch.36

… ►

36.10.4 $$\frac{{\partial }^3{\mathrm{\Psi }}_2}{{\partial x}^3}-\frac{1}{2}y\frac{\partial {\mathrm{\Psi }}_2}{\partial x}-\frac{\mathrm{i}}{4}x{\mathrm{\Psi }}_2=0.$$

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Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $y$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §36.10(i), §36.10(i), §36.10 and Ch.36

… ►In terms of the normal forms (36.2.2) and (36.2.3), the ${\mathrm{\Psi }}^{(\mathrm{U})}\left(𝐱\right)$ satisfy the following operator equations …

5: 36.3 Visualizations of Canonical Integrals

… ►

…

Figure 36.3.1: Modulus of Pearcey integral $|{\mathrm{\Psi }}_2\left(x,y\right)|$. …

►

…

Figure 36.3.2: Modulus of swallowtail canonical integral function $|{\mathrm{\Psi }}_3\left(x,y,3\right)|$. …

… ►In Figure 36.3.13(a) points of confluence of phase contours are zeros of ${\mathrm{\Psi }}_2\left(x,y\right)$; similarly for other contour plots in this subsection. In Figure 36.3.13(b) points of confluence of all colors are zeros of ${\mathrm{\Psi }}_2\left(x,y\right)$; similarly for other density plots in this subsection. ►

…

Figure 36.3.13: Phase of Pearcey integral $\mathrm{ph}{\mathrm{\Psi }}_2\left(x,y\right)$. …

…

6: 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). …

7: 5.24 Software

… ►

§5.24(iii) $\psi \left(x\right)$, ${\psi }^{(n)}\left(x\right)$, $x\in ℝ$

… ►

§5.24(iv) $\mathrm{\Gamma }\left(z\right)$, $\psi \left(z\right)$, ${\psi }^{(n)}\left(z\right)$, $z\in ℂ$

…

8: 19.11 Addition Theorems

… ►In the case of $\theta ,\varphi \in [0,\pi /2)$ and , we can use … ►

§19.11(ii) Case $\psi =\pi /2$

… ►

$$\sin \theta =(\sin \psi )/\sqrt{(1+\cos \psi )(1+\mathrm{\Delta }(\psi ))},$$

►

$$\cos \theta =\sqrt{\frac{(\cos \psi )+\mathrm{\Delta }(\psi )}{1+\mathrm{\Delta }(\psi )}},$$

►

$$\tan \theta =\tan \left(\frac{1}{2}\psi \right)\sqrt{\frac{1+\cos \psi }{(\cos \psi )+\mathrm{\Delta }(\psi )}},$$

…

9: 5.22 Tables

… ►Abramowitz and Stegun (1964, Chapter 6) tabulates $\mathrm{\Gamma }\left(x\right)$, $\ln \mathrm{\Gamma }\left(x\right)$, $\psi \left(x\right)$, and ${\psi }^{\prime }\left(x\right)$ for $x=1(.005)2$ to 10D; ${\psi }^{\prime \prime }\left(x\right)$ and ${\psi }^{(3)}\left(x\right)$ for $x=1(.01)2$ to 10D; $\mathrm{\Gamma }\left(n\right)$, $1/\mathrm{\Gamma }\left(n\right)$, $\mathrm{\Gamma }\left(n+\frac{1}{2}\right)$, $\psi \left(n\right)$, ${\log }_{10}\mathrm{\Gamma }\left(n\right)$, ${\log }_{10}\mathrm{\Gamma }\left(n+\frac{1}{3}\right)$, ${\log }_{10}\mathrm{\Gamma }\left(n+\frac{1}{2}\right)$, and ${\log }_{10}\mathrm{\Gamma }\left(n+\frac{2}{3}\right)$ for $n=1(1)101$ to 8–11S; $\mathrm{\Gamma }\left(n+1\right)$ for $n=100(100)1000$ to 20S. Zhang and Jin (1996, pp. 67–69 and 72) tabulates $\mathrm{\Gamma }\left(x\right)$, $1/\mathrm{\Gamma }\left(x\right)$, $\mathrm{\Gamma }\left(-x\right)$, $\ln \mathrm{\Gamma }\left(x\right)$, $\psi \left(x\right)$, $\psi \left(-x\right)$, ${\psi }^{\prime }\left(x\right)$, and ${\psi }^{\prime }\left(-x\right)$ for $x=0(.1)5$ to 8D or 8S; $\mathrm{\Gamma }\left(n+1\right)$ for $n=0(1)100(10)250(50)500(100)3000$ to 51S. … ►This reference also includes $\psi \left(x+\mathrm{i}y\right)$ for the same arguments to 5D. Zhang and Jin (1996, pp. 70, 71, and 73) tabulates the real and imaginary parts of $\mathrm{\Gamma }\left(x+\mathrm{i}y\right)$, $\ln \mathrm{\Gamma }\left(x+\mathrm{i}y\right)$, and $\psi \left(x+\mathrm{i}y\right)$ for $x=0.5,1,5,10$, $y=0(.5)10$ to 8S.

10: 36.1 Special Notation

… ►The main functions covered in this chapter are cuspoid catastrophes ${\mathrm{\Phi }}_K(t;𝐱)$; umbilic catastrophes with codimension three ${\mathrm{\Phi }}^{(\mathrm{E})}(s,t;𝐱)$, ${\mathrm{\Phi }}^{(\mathrm{H})}(s,t;𝐱)$; canonical integrals ${\mathrm{\Psi }}_K\left(𝐱\right)$, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$, ${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$; diffraction catastrophes ${\mathrm{\Psi }}_K(𝐱;k)$, ${\mathrm{\Psi }}^{(\mathrm{E})}(𝐱;k)$, ${\mathrm{\Psi }}^{(\mathrm{H})}(𝐱;k)$ generated by the catastrophes. …