重庆时时彩近120期【AG真人官方qee9.com】vec

SearchAdvancedHelp

(0.002 seconds)

11—20 of 256 matching pages

11: 10.30 Limiting Forms

… ►

§10.30(i) $z\to 0$

►When $\nu$ is fixed and $z\to 0$, …For $K_{\nu }\left(x\right)$, when $\nu$ is purely imaginary and $x\to 0+$, see (10.45.2) and (10.45.7). ►

§10.30(ii) $z\to \mathrm{\infty }$

►When $\nu$ is fixed and $z\to \mathrm{\infty }$, …

12: 18.24 Hahn Class: Asymptotic Approximations

… ►When the parameters $\alpha$ and $\beta$ are fixed and the ratio $n/N=c$ is a constant in the interval (0,1), uniform asymptotic formulas (as $n\to \mathrm{\infty }$ ) of the Hahn polynomials $Q_n(z;\alpha ,\beta ,N)$ can be found in Lin and Wong (2013) for $z$ in three overlapping regions, which together cover the entire complex plane. … ►With $x=\lambda N$ and $\nu =n/N$, Li and Wong (2000) gives an asymptotic expansion for $K_n(x;p,N)$ as $n\to \mathrm{\infty }$, that holds uniformly for $\lambda$ and $\nu$ in compact subintervals of $(0,1)$. … ►For two asymptotic expansions of $M_n(nx;\beta ,c)$ as $n\to \mathrm{\infty }$, with $\beta$ and $c$ fixed, see Jin and Wong (1998) and Wang and Wong (2011). … ►Dunster (2001b) provides various asymptotic expansions for $C_n(x;a)$ as $n\to \mathrm{\infty }$, in terms of elementary functions or in terms of Bessel functions. … ►For an asymptotic expansion of $P_n^{(\lambda )}(nx;\varphi )$ as $n\to \mathrm{\infty }$, with $\varphi$ fixed, see Li and Wong (2001). …

13: 24.11 Asymptotic Approximations

… ►As $n\to \mathrm{\infty }$ ►

24.11.1 $${(-1)}^{n+1}{B}_{2n}\sim \frac{2(2n)!}{{(2\pi )}^{2n}},$$

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$) and $n$: integer

Referenced by:

§24.11

Permalink:

http://dlmf.nist.gov/24.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §24.11 and Ch.24

►

24.11.2 $${(-1)}^{n+1}{B}_{2n}\sim 4\sqrt{\pi n}{\left(\frac{n}{\pi \mathrm{e}}\right)}^{2n},$$

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $n$: integer

Permalink:

http://dlmf.nist.gov/24.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §24.11 and Ch.24

►

24.11.3 $${(-1)}^n{E}_{2n}\sim \frac{2^{2n+2}(2n)!}{{\pi }^{2n+1}},$$

ⓘ

Symbols:

$E_n$: Euler numbers, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$) and $n$: integer

Permalink:

http://dlmf.nist.gov/24.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §24.11 and Ch.24

►

24.11.4 $${(-1)}^n{E}_{2n}\sim 8\sqrt{\frac{n}{\pi }}{\left(\frac{4n}{\pi \mathrm{e}}\right)}^{2n}.$$

ⓘ

Symbols:

$E_n$: Euler numbers, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $n$: integer

Referenced by:

§24.11

Permalink:

http://dlmf.nist.gov/24.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §24.11 and Ch.24

…

14: 18.21 Hahn Class: Interrelations

… ►

Hahn $\to$ Krawtchouk

… ►

Hahn $\to$ Meixner

… ►

Krawtchouk $\to$ Charlier

… ►

Meixner $\to$ Charlier

… ►

Continuous Hahn $\to$ Meixner–Pollaczek

…

15: 4.31 Special Values and Limits

… ►

4.31.1 $$\underset{z\to 0}{lim}\frac{\sinh z}{z}=1,$$

ⓘ

Symbols:

$\sinh z$: hyperbolic sine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.31.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.31 and Ch.4

►

4.31.2 $$\underset{z\to 0}{lim}\frac{\tanh z}{z}=1,$$

ⓘ

Symbols:

$\tanh z$: hyperbolic tangent function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.31.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.31 and Ch.4

►

4.31.3 $$\underset{z\to 0}{lim}\frac{\cosh z-1}{z^2}=\frac{1}{2}.$$

ⓘ

Symbols:

$\cosh z$: hyperbolic cosine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.31.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.31 and Ch.4

16: 28.9 Zeros

… ►For $q\to \mathrm{\infty }$ the zeros of ${\mathrm{ce}}_{2n}(z,q)$ and ${\mathrm{se}}_{2n+1}(z,q)$ approach asymptotically the zeros of ${\mathrm{𝐻𝑒}}_{2n}\left(q^{1/4}(\pi -2z)\right)$, and the zeros of ${\mathrm{ce}}_{2n+1}(z,q)$ and ${\mathrm{se}}_{2n+2}(z,q)$ approach asymptotically the zeros of ${\mathrm{𝐻𝑒}}_{2n+1}\left(q^{1/4}(\pi -2z)\right)$. …Furthermore, for $q>0$ ${\mathrm{ce}}_m(z,q)$ and ${\mathrm{se}}_m(z,q)$ also have purely imaginary zeros that correspond uniquely to the purely imaginary $z$-zeros of $J_m\left(2\sqrt{q}\cos z\right)$ (§10.21(i)), and they are asymptotically equal as $q\to 0$ and $\left|\mathrm{\Im }z\right|\to \mathrm{\infty }$. …

17: 19.27 Asymptotic Approximations and Expansions

… ►

19.27.2 $$R_F(x,y,z)=\frac{1}{2\sqrt{z}}\left(\ln \frac{8z}{a+g}\right)\left(1+O\left(\frac{a}{z}\right)\right),$$ $a/z\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\ln z$: principal branch of logarithm function, $a$ and $g$

Referenced by:

§19.20(iii), §19.27(ii)

Permalink:

http://dlmf.nist.gov/19.27.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(ii), §19.27 and Ch.19

►

19.27.3 $$R_F(x,y,z)=R_F(0,y,z)-\frac{1}{\sqrt{h}}\left(\sqrt{\frac{x}{h}}+O\left(\frac{x}{h}\right)\right),$$ $x/h\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_F(x,y,z)$: symmetric elliptic integral of first kind and $h$

Permalink:

http://dlmf.nist.gov/19.27.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(ii), §19.27 and Ch.19

… ►

19.27.4 $$R_G(x,y,z)=\frac{\sqrt{z}}{2}\left(1+O\left(\frac{a}{z}\ln \frac{z}{a}\right)\right),$$ $a/z\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_G(x,y,z)$: symmetric elliptic integral of second kind, $\ln z$: principal branch of logarithm function and $a$

Permalink:

http://dlmf.nist.gov/19.27.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(iii), §19.27 and Ch.19

►

19.27.5 $$R_G(x,y,z)=R_G(0,y,z)+\sqrt{x}O\left(\sqrt{x/h}\right),$$ $x/h\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_G(x,y,z)$: symmetric elliptic integral of second kind and $h$

Permalink:

http://dlmf.nist.gov/19.27.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(iii), §19.27 and Ch.19

►

19.27.6 $$R_G(0,y,z)=\frac{\sqrt{z}}{2}+\frac{y}{8\sqrt{z}}\left(\ln \left(\frac{16z}{y}\right)-1\right)\left(1+O\left(\frac{y}{z}\right)\right),$$ $y/z\to 0$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $R_G(x,y,z)$: symmetric elliptic integral of second kind and $\ln z$: principal branch of logarithm function

Permalink:

http://dlmf.nist.gov/19.27.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(iii), §19.27 and Ch.19

…

18: 36.11 Leading-Order Asymptotics

… ►

36.11.3

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $o\left(x\right)$: order less than and $y$: real parameter

Permalink:

http://dlmf.nist.gov/36.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §36.11, §36.11 and Ch.36

►

36.11.4

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\exp z$: exponential function, $o\left(x\right)$: order less than and $x$: real parameter

Referenced by:

§36.11

Permalink:

http://dlmf.nist.gov/36.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §36.11, §36.11 and Ch.36

►

36.11.5 $${\mathrm{\Psi }}_3\left(0,y,0\right)=\overline{{\mathrm{\Psi }}_3(0,-y,0)}=\exp \left(\frac{1}{4}\mathrm{i}\pi \right)\sqrt{\pi /y}\left(1-(\mathrm{i}/\sqrt{3})\exp \left(\frac{3}{2}\mathrm{i}{(2y/5)}^{5/3}\right)+o\left(1\right)\right),$$ $y\to +\mathrm{\infty }$.

ⓘ

Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $o\left(x\right)$: order less than and $y$: real parameter

Permalink:

http://dlmf.nist.gov/36.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §36.11, §36.11 and Ch.36

►

36.11.6

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $o\left(x\right)$: order less than and $z$: real parameter

Permalink:

http://dlmf.nist.gov/36.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §36.11, §36.11 and Ch.36

►

36.11.7 $${\mathrm{\Psi }}^{(\mathrm{E})}\left(0,0,z\right)=\frac{\pi }{z}\left(\mathrm{i}+\sqrt{3}\exp \left(\frac{4}{27}\mathrm{i}{z}^3\right)+o\left(1\right)\right),$$ $z\to \pm \mathrm{\infty }$,

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $o\left(x\right)$: order less than and $z$: real parameter

Permalink:

http://dlmf.nist.gov/36.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §36.11, §36.11 and Ch.36

…

19: 11.11 Asymptotic Expansions of Anger–Weber Functions

… ►Then as $z\to \mathrm{\infty }$ in $|\mathrm{ph}z|\le \pi -\delta$ … ►If $z$ is fixed, and $\nu \to \mathrm{\infty }$ in $|\mathrm{ph}\nu |\le \pi$ in such a way that $\nu$ is bounded away from the set of all integers, then … ►as $n\to \pm \mathrm{\infty }$. … ►In particular, as $\nu \to \mathrm{\infty }$, … ►Also, as $\nu \to \mathrm{\infty }$ in $|\mathrm{ph}\nu |\le 2\pi -\delta$, …

20: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series

… ►

22.12.4 $$2\mathrm{i}K\mathrm{dn}(2Kt,k)=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\frac{\pi }{\tan \left(\pi (t-(n+\frac{1}{2})\tau )\right)}=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\left(\underset{M\to \mathrm{\infty }}{lim}\sum _{m=-M}^M\frac{1}{t-m-(n+\frac{1}{2})\tau }\right).$$

ⓘ

Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $\tan z$: tangent function, $k$: modulus and $\tau$: ratio

Referenced by:

§22.12, §22.12

Permalink:

http://dlmf.nist.gov/22.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22

… ►

22.12.7 $$2\mathrm{i}K{k}^{\prime }\mathrm{nd}(2Kt,k)=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\frac{\pi }{\tan \left(\pi (t+\frac{1}{2}-(n+\frac{1}{2})\tau )\right)}=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\underset{M\to \mathrm{\infty }}{lim}\left(\sum _{m=-M}^M\frac{1}{t+\frac{1}{2}-m-(n+\frac{1}{2})\tau }\right),$$

ⓘ

Symbols:

$\mathrm{nd}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $\tan z$: tangent function, $k$: modulus, $k^{\prime }$: complementary modulus and $\tau$: ratio

Permalink:

http://dlmf.nist.gov/22.12.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22

… ►

22.12.10 $$-2K{k}^{\prime }\mathrm{sc}(2Kt,k)=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\frac{\pi }{\tan \left(\pi (t+\frac{1}{2}-n\tau )\right)}=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\left(\underset{M\to \mathrm{\infty }}{lim}\sum _{m=-M}^M\frac{1}{t+\frac{1}{2}-m-n\tau }\right),$$

ⓘ

Symbols:

$\mathrm{sc}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\tan z$: tangent function, $k$: modulus, $k^{\prime }$: complementary modulus and $\tau$: ratio

Permalink:

http://dlmf.nist.gov/22.12.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22

… ►

22.12.13 $$2K\mathrm{cs}(2Kt,k)=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\frac{\pi }{\tan \left(\pi (t-n\tau )\right)}=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\left(\underset{M\to \mathrm{\infty }}{lim}\sum _{m=-M}^M\frac{1}{t-m-n\tau }\right).$$

ⓘ

Symbols:

$\mathrm{cs}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\tan z$: tangent function, $k$: modulus and $\tau$: ratio

Referenced by:

§22.12

Permalink:

http://dlmf.nist.gov/22.12.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22