韩国顺天乡大学在读证明办理《做证微fuk7778》neq

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11: 5.5 Functional Relations

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5.5.3 $$\mathrm{\Gamma }\left(z\right)\mathrm{\Gamma }\left(1-z\right)=\pi /\sin \left(\pi z\right),$$ $z\ne 0,\pm 1,\mathrm{\dots }$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function and $z$: complex variable

A&S Ref:

6.1.17 (without the condition on $z$.)

Referenced by:

§10.19(i), §15.4(ii), §15.8(ii), (25.9.2), §5.21, (9.10.18), (9.12.15)

Permalink:

http://dlmf.nist.gov/5.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.5(ii), §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

… ►For $2z\ne 0,-1,-2,\mathrm{\dots }$, … ►For $nz\ne 0,-1,-2,\mathrm{\dots }$, …

12: 15.15 Sums

… ►Here $z_0$ ($\ne 0$) is an arbitrary complex constant and the expansion converges when $|z-z_0|>\max (|z_0|,|z_0-1|)$. …

13: 4.25 Continued Fractions

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4.25.1 $$\tan z=\frac{z}{1-}\frac{z^2}{3-}\frac{z^2}{5-}\frac{z^2}{7-}\mathrm{\cdots },$$ $z\ne \pm \frac{1}{2}\pi$, $\pm \frac{3}{2}\pi$, $\mathrm{\dots }$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\tan z$: tangent function and $z$: complex variable

A&S Ref:

4.3.94

Permalink:

http://dlmf.nist.gov/4.25.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.25 and Ch.4

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4.25.2 $$\tan \left(az\right)=\frac{a\tan z}{1+}\frac{(1-a^2){\tan }^{2}z}{3+}\frac{(4-a^2){\tan }^{2}z}{5+}\frac{(9-a^2){\tan }^{2}z}{7+}\mathrm{\cdots },$$ , $az\ne \pm \frac{1}{2}\pi ,\pm \frac{3}{2}\pi ,\mathrm{\dots }$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Re }$: real part, $\tan z$: tangent function, $a$: real or complex constant and $z$: complex variable

A&S Ref:

4.3.95

Permalink:

http://dlmf.nist.gov/4.25.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.25 and Ch.4

…

14: 5.8 Infinite Products

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5.8.1 $$\mathrm{\Gamma }\left(z\right)=\underset{k\to \mathrm{\infty }}{lim}\frac{k!k^z}{z(z+1)\mathrm{\cdots }(z+k)},$$ $z\ne 0,-1,-2,\mathrm{\dots }$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $!$: factorial (as in $n!$), $k$: nonnegative integer and $z$: complex variable

A&S Ref:

6.1.2

Referenced by:

§5.8

Permalink:

http://dlmf.nist.gov/5.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §5.8 and Ch.5

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5.8.3 $${\left|\frac{\mathrm{\Gamma }\left(x\right)}{\mathrm{\Gamma }\left(x+\mathrm{i}y\right)}\right|}^2=\prod _{k=0}^{\mathrm{\infty }}\left(1+\frac{y^2}{{(x+k)}^2}\right),$$ $x\ne 0,-1,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $x$: real variable and $y$: real variable

A&S Ref:

6.1.25 (where the formula for the reciprocal is given.)

Referenced by:

§5.8

Permalink:

http://dlmf.nist.gov/5.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.8 and Ch.5

…

15: 13.17 Continued Fractions

… ►If $\kappa ,\mu \in ℂ$ such that $\mu \pm (\kappa -\frac{1}{2})\ne -1,-2,-3,\mathrm{\dots }$, then … ►If $\kappa ,\mu \in ℂ$ such that $\mu +\frac{1}{2}\pm (\kappa +1)\ne -1,-2,-3,\mathrm{\dots }$, then …

16: 4.5 Inequalities

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4.5.1 $x>-1$, $x\ne 0$,

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Symbols:

$\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

4.1.33

Referenced by:

§4.5(i), §4.5(ii)

Permalink:

http://dlmf.nist.gov/4.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.5(i), §4.5 and Ch.4

►

4.5.2 , $x\ne 0$,

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Symbols:

$\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

4.1.34

Referenced by:

§4.5(i), §4.5(ii)

Permalink:

http://dlmf.nist.gov/4.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.5(i), §4.5 and Ch.4

… ►In (4.5.7)–(4.5.12) it is assumed that $x\ne 0$. …

17: 10.13 Other Differential Equations

… ►In the following equations $\nu ,\lambda ,p,q$, and $r$ are real or complex constants with $\lambda \ne 0$, $p\ne 0$, and $q\ne 0$. …

18: 14.17 Integrals

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14.17.2 $$\int {\left(1-x^2\right)}^{\mu /2}{𝖯}_{\nu }^{\mu }\left(x\right)dx=\frac{{\left(1-x^2\right)}^{(\mu +1)/2}}{(\nu -\mu )(\nu +\mu +1)}{𝖯}_{\nu }^{\mu +1}\left(x\right),$$ $\mu \ne \nu$ or $-\nu -1$.

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Symbols:

: Ferrers function of the first kind, $dx$: differential of $x$ , $\int$: integral, $x$: real variable, $\mu$: general order and $\nu$: general degree

Permalink:

http://dlmf.nist.gov/14.17.E2

Encodings:

pMML, png, TeX

See also:

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

… ►

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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14.17.12 $${\int }_{-1}^1{𝖰}_{\nu }\left(x\right){𝖰}_{\lambda }\left(x\right)dx=\frac{\left((\psi \left(\nu +1\right)-\psi \left(\lambda +1\right))(1+\cos \left(\nu \pi \right)\cos \left(\lambda \pi \right))+\frac{1}{2}\pi \sin \left((\lambda -\nu )\pi \right)\right)}{(\lambda -\nu )(\lambda +\nu +1)},$$ $\lambda \ne \nu$ or $-\nu -1$, $\lambda \text{ and }\nu \ne -1,-2,-3,\mathrm{\dots }$.

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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, $\sin z$: sine function, $x$: real variable and $\nu$: general degree

A&S Ref:

8.14.6

Permalink:

http://dlmf.nist.gov/14.17.E12

Encodings:

pMML, png, TeX

See also:

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

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14.17.16 $${\int }_{-1}^1{𝖯}_l^m\left(x\right){𝖰}_n^m\left(x\right)dx=\frac{\left(1-{(-1)}^{l+n}\right)(l+m)!}{(l-n)(l+n+1)(l-m)!},$$ $l,m,n=0,1,2,\mathrm{\dots }$, $l\ne n$.

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Symbols:

: Ferrers function of the first kind, : Ferrers function of the second kind, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $x$: real variable, $m$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

8.14.10 (corrected)

Referenced by:

§14.17(iv)

Permalink:

http://dlmf.nist.gov/14.17.E16

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

…

19: 25.10 Zeros

… ►The product representation (25.2.11) implies $\zeta \left(s\right)\ne 0$ for $\mathrm{\Re }s>1$. Also, $\zeta \left(s\right)\ne 0$ for $\mathrm{\Re }s=1$, a property first established in Hadamard (1896) and de la Vallée Poussin (1896a, b) in the proof of the prime number theorem (25.16.3). …Except for the trivial zeros, $\zeta \left(s\right)\ne 0$ for $\mathrm{\Re }s\le 0$. …

20: 4.22 Infinite Products and Partial Fractions

… ►When $z\ne n\pi$, $n\in \mathbb{Z}$, …