韩国顺天乡大学在读证明办理《做证微fuk7778》neq

SearchAdvancedHelp

The term"fuk7778" was not found.Possible alternative term: "fukui".

(0.003 seconds)

1—10 of 222 matching pages

1: 10.10 Continued Fractions

… ►Assume $J_{\nu -1}\left(z\right)\ne 0$. … ►

10.10.1 $$\frac{J_{\nu }\left(z\right)}{J_{\nu -1}\left(z\right)}=\frac{1}{2\nu z^{-1}-}\frac{1}{2(\nu +1)z^{-1}-}\frac{1}{2(\nu +2)z^{-1}-}\mathrm{\cdots },$$ $z\ne 0$,

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.73

Referenced by:

§10.33, §10.74(v)

Permalink:

http://dlmf.nist.gov/10.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.10 and Ch.10

►

10.10.2 $$\frac{J_{\nu }\left(z\right)}{J_{\nu -1}\left(z\right)}=\frac{\frac{1}{2}z/\nu }{1-}\frac{\frac{1}{4}{z}^2/(\nu (\nu +1))}{1-}\frac{\frac{1}{4}{z}^2/((\nu +1)(\nu +2))}{1-}\mathrm{\cdots },$$ $\nu \ne 0,-1,-2,\mathrm{\dots }$.

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.73

Referenced by:

§10.33, §10.74(v)

Permalink:

http://dlmf.nist.gov/10.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.10 and Ch.10

…

2: 10.33 Continued Fractions

… ►Assume $I_{\nu -1}\left(z\right)\ne 0$. … ►

10.33.1 $$\frac{I_{\nu }\left(z\right)}{I_{\nu -1}\left(z\right)}=\frac{1}{2\nu z^{-1}+}\frac{1}{2(\nu +1)z^{-1}+}\frac{1}{2(\nu +2)z^{-1}+}\mathrm{\cdots },$$ $z\ne 0$,

ⓘ

Symbols:

$I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.74(v)

Permalink:

http://dlmf.nist.gov/10.33.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.33 and Ch.10

►

10.33.2 $$\frac{I_{\nu }\left(z\right)}{I_{\nu -1}\left(z\right)}=\frac{\frac{1}{2}z/\nu }{1+}\frac{\frac{1}{4}{z}^2/(\nu (\nu +1))}{1+}\frac{\frac{1}{4}{z}^2/((\nu +1)(\nu +2))}{1+}\mathrm{\cdots },$$ $\nu \ne 0,-1,-2,\mathrm{\dots }$.

ⓘ

Symbols:

$I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.74(v)

Permalink:

http://dlmf.nist.gov/10.33.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.33 and Ch.10

…

3: 4.10 Integrals

… ►

4.10.3 $$\int z^n\ln zdz=\frac{z^{n+1}}{n+1}\ln z-\frac{z^{n+1}}{{(n+1)}^2},$$ $n\ne -1$,

ⓘ

Symbols:

$dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, $n$: integer and $z$: complex variable

A&S Ref:

4.1.50

Permalink:

http://dlmf.nist.gov/4.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.10(i), §4.10 and Ch.4

… ►For $a,b\ne 0$, ►

4.10.8 $$\int {\mathrm{e}}^{az}dz=\frac{{\mathrm{e}}^{az}}{a},$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $z$: complex variable and $a\ne 0$: complex

A&S Ref:

4.2.54

Referenced by:

§4.10(ii)

Permalink:

http://dlmf.nist.gov/4.10.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §4.10(ii), §4.10 and Ch.4

►

4.10.9 $$\int \frac{dz}{{\mathrm{e}}^{az}+b}=\frac{1}{ab}(az-\ln \left({\mathrm{e}}^{az}+b\right)),$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\ln z$: principal branch of logarithm function, $z$: complex variable, $a\ne 0$: complex and $b\ne 0$: complex

Permalink:

http://dlmf.nist.gov/4.10.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §4.10(ii), §4.10 and Ch.4

►

4.10.10 $$\int \frac{{\mathrm{e}}^{az}-1}{{\mathrm{e}}^{az}+1}dz=\frac{2}{a}\ln \left({\mathrm{e}}^{az/2}+{\mathrm{e}}^{-az/2}\right),$$

ⓘ

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\ln z$: principal branch of logarithm function, $z$: complex variable and $a\ne 0$: complex

Referenced by:

§4.10(ii)

Permalink:

http://dlmf.nist.gov/4.10.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §4.10(ii), §4.10 and Ch.4

…

4: 13.5 Continued Fractions

… ►If $a,b\in ℂ$ such that $a\ne -1,-2,-3,\mathrm{\dots }$, and $a-b\ne 0,1,2,\mathrm{\dots }$, then … ►If $a,b\in ℂ$ such that $a\ne 0,-1,-2,\mathrm{\dots }$, and $b-a\ne 2,3,4,\mathrm{\dots }$, then …

5: 4.6 Power Series

… ►

4.6.1 $$\ln \left(1+z\right)=z-\frac{1}{2}{z}^2+\frac{1}{3}{z}^3-\mathrm{\cdots },$$ $|z|\le 1$, $z\ne -1$,

ⓘ

Symbols:

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

A&S Ref:

4.1.24

Referenced by:

§4.45(i), §4.6(i), (8.15.2)

Permalink:

http://dlmf.nist.gov/4.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.6(i), §4.6 and Ch.4

… ►

4.6.3 $$\ln z=(z-1)-\frac{1}{2}{(z-1)}^2+\frac{1}{3}{(z-1)}^3-\mathrm{\cdots },$$ $|z-1|\le 1$, $z\ne 0$,

ⓘ

Symbols:

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

A&S Ref:

4.1.26

Permalink:

http://dlmf.nist.gov/4.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.6(i), §4.6 and Ch.4

►

4.6.4 $$\ln z=2\left(\left(\frac{z-1}{z+1}\right)+\frac{1}{3}{\left(\frac{z-1}{z+1}\right)}^3+\frac{1}{5}{\left(\frac{z-1}{z+1}\right)}^5+\mathrm{\cdots }\right),$$ $\mathrm{\Re }z\ge 0$, $z\ne 0$,

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

4.1.27

Permalink:

http://dlmf.nist.gov/4.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.6(i), §4.6 and Ch.4

►

4.6.5 $$\ln \left(\frac{z+1}{z-1}\right)=2\left(\frac{1}{z}+\frac{1}{3z^3}+\frac{1}{5z^5}+\mathrm{\cdots }\right),$$ $|z|\ge 1$, $z\ne \pm 1$,

ⓘ

Symbols:

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

A&S Ref:

4.1.28

Permalink:

http://dlmf.nist.gov/4.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.6(i), §4.6 and Ch.4

►

4.6.6 $$\ln \left(z+a\right)=\ln a+2\left(\left(\frac{z}{2a+z}\right)+\frac{1}{3}{\left(\frac{z}{2a+z}\right)}^3+\frac{1}{5}{\left(\frac{z}{2a+z}\right)}^5+\mathrm{\cdots }\right),$$ $a>0$, $\mathrm{\Re }z\ge -a$, $z\ne -a$.

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part, $a$: real or complex constant and $z$: complex variable

A&S Ref:

4.1.29

Permalink:

http://dlmf.nist.gov/4.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.6(i), §4.6 and Ch.4

…

6: 25.15 Dirichlet $L$-functions

… ►If $\chi \ne {\chi }_1$, then $L(s,\chi )$ is an entire function of $s$. …This implies that $L(s,\chi )\ne 0$ if $\mathrm{\Re }s>1$. ►Equations (25.15.3) and (25.15.4) hold for all $s$ if $\chi \ne {\chi }_1$, and for all $s$ ($\ne 1$) if $\chi ={\chi }_1$: … ►Since $L(s,\chi )\ne 0$ if $\mathrm{\Re }s>1$, (25.15.5) shows that for a primitive character $\chi$ the only zeros of $L(s,\chi )$ for (the so-called trivial zeros) are as follows: … ►

25.15.9 $$L(1,\chi )\ne 0\text{ if }\chi \ne {\chi }_1,$$

ⓘ

Symbols:

$L(s,\chi )$: Dirichlet $L$-function and $\chi (n)$: Dirichlet character

Keywords:

zeros

Source:

Apostol (1976, p. 149)

Permalink:

http://dlmf.nist.gov/25.15.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §25.15(ii), §25.15 and Ch.25

…

7: 28.17 Stability as $x\to \pm \mathrm{\infty }$

… ►However, if $\mathrm{\Im }\nu \ne 0$, then $(a,q)$ always comprises an unstable pair. … ►For real $a$ and $q$ $(\ne 0)$ the stable regions are the open regions indicated in color in Figure 28.17.1. …

8: 33.18 Limiting Forms for Large $\mathrm{\ell }$

… ►As $\mathrm{\ell }\to \mathrm{\infty }$ with $ϵ$ and $r$ ($\ne 0$) fixed, …

9: 14.8 Behavior at Singularities

… ►

14.8.1 $${𝖯}_{\nu }^{\mu }\left(x\right)\sim \frac{1}{\mathrm{\Gamma }\left(1-\mu \right)}{\left(\frac{2}{1-x}\right)}^{\mu /2},$$ $\mu \ne 1,2,3,\mathrm{\dots }$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $\sim$: asymptotic equality, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.30(ii), §14.8(i)

Permalink:

http://dlmf.nist.gov/14.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.8(i), §14.8 and Ch.14

►

14.8.2 $${𝖯}_{\nu }^m\left(x\right)\sim {(-1)}^m\frac{{\left(\nu -m+1\right)}_{2m}}{m!}{\left(\frac{1-x}{2}\right)}^{m/2},$$ $m=1,2,3,\mathrm{\dots }$, $\nu \ne m-1,m-2,\mathrm{\dots },-m$,

ⓘ

Symbols:

: Ferrers function of the first kind, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: asymptotic equality, $!$: factorial (as in $n!$), $x$: real variable, $m$: nonnegative integer and $\nu$: general degree

Referenced by:

§14.30(ii), §14.8(i)

Permalink:

http://dlmf.nist.gov/14.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.8(i), §14.8 and Ch.14

… ►

14.8.4 $${𝖰}_{\nu }^{\mu }\left(x\right)\sim \frac{1}{2}\cos \left(\mu \pi \right)\mathrm{\Gamma }\left(\mu \right){\left(\frac{2}{1-x}\right)}^{\mu /2},$$ $\mu \ne \frac{1}{2},\frac{3}{2},\frac{5}{2},\mathrm{\dots }$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the second kind, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.8(i)

Permalink:

http://dlmf.nist.gov/14.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.8(i), §14.8 and Ch.14

… ►

14.8.7 $$P_{\nu }^{\mu }\left(x\right)\sim \frac{1}{\mathrm{\Gamma }\left(1-\mu \right)}{\left(\frac{2}{x-1}\right)}^{\mu /2},$$ $\mu \ne 1,2,3,\mathrm{\dots }$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\sim$: asymptotic equality, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.21(iii), §14.24, §14.8(ii)

Permalink:

http://dlmf.nist.gov/14.8.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §14.8(ii), §14.8 and Ch.14

►

14.8.8 $$P_{\nu }^m\left(x\right)\sim \frac{\mathrm{\Gamma }\left(\nu +m+1\right)}{m!\mathrm{\Gamma }\left(\nu -m+1\right)}{\left(\frac{x-1}{2}\right)}^{m/2},$$ $m=1,2,3,\mathrm{\dots }$, $\nu \pm m\ne -1,-2,-3,\mathrm{\dots }$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\sim$: asymptotic equality, $!$: factorial (as in $n!$), $x$: real variable, $m$: nonnegative integer and $\nu$: general degree

Permalink:

http://dlmf.nist.gov/14.8.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §14.8(ii), §14.8 and Ch.14

…

10: 4.8 Identities

… ►In (4.8.1)–(4.8.4) $z_1{z}_2\ne 0$. … ►In (4.8.5)–(4.8.7) and (4.8.10) $z\ne 0$. … ►If $a\ne 0$ and $a^z$ has its general value, then …If $a\ne 0$ and $a^z$ has its principal value, then … ►

4.8.14 $$a^{z_1}{a}^{z_2}=a^{z_1+z_2},$$ $a\ne 0$,

ⓘ

Symbols:

$a$: real or complex constant and $z$: complex variable

A&S Ref:

4.2.15

Referenced by:

Erratum (V1.0.20) for Equation (4.8.14)

Permalink:

http://dlmf.nist.gov/4.8.E14

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.20):

The constraint $a\ne 0$ was added.

Suggested 2018-08-13 by Ted Ersek

See also:

Annotations for §4.8(ii), §4.8 and Ch.4

…