莱斯布里奇大学学历认证【购证 微kaa77788】big

SearchAdvancedHelp

(0.002 seconds)

11—20 of 93 matching pages

11: 28.14 Fourier Series

… ►

28.14.6 $$\frac{c_{2m}^{\nu }(q)}{c_{2m\mp 2}^{\nu }(q)}=\frac{-q}{4m^2}\left(1+O\left(\frac{1}{m}\right)\right),$$ $m\to \pm \mathrm{\infty }$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $m$: integer, $q=h^2$: parameter, $\nu$: complex parameter and $c_{2m}(q)$: coefficients

Permalink:

http://dlmf.nist.gov/28.14.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §28.14 and Ch.28

… ►

28.14.10 $$c_{2m}^{\nu }(q)=\left(\frac{{(-1)}^m{q}^m\mathrm{\Gamma }\left(\nu +1\right)}{m!{\mathrm{\hspace{0.17em}2}}^{2m}\mathrm{\Gamma }\left(\nu +m+1\right)}+O\left(q^{m+2}\right)\right)c_0^{\nu }(q).$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $!$: factorial (as in $n!$), $m$: integer, $q=h^2$: parameter, $\nu$: complex parameter and $c_{2m}(q)$: coefficients

A&S Ref:

20.3.16

Permalink:

http://dlmf.nist.gov/28.14.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §28.14 and Ch.28

12: 18.27 $q$-Hahn Class

… ►The generic (top level) cases are the $q$-Hahn polynomials and the big $q$-Jacobi polynomials, each of which depends on three further parameters. … ►

§18.27(iii) Big $q$-Jacobi Polynomials

… ►

18.27.10 $$p_n(x)=P_n^{(\alpha ,\beta )}(x;c,d;q)$$

ⓘ

Symbols:

$P_n^{(\alpha ,\beta )}(x;c,d;q)$: big $q$-Jacobi polynomial, $p_n(x)$: polynomial of degree $n$ , $q$: real variable, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.27.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §18.27(iii), §18.27 and Ch.18

… ►

From Big $q$-Jacobi to Jacobi

… ►

From Big $q$-Jacobi to Little $q$-Jacobi

…

13: 13.20 Uniform Asymptotic Approximations for Large $\mu$

… ►

13.20.1 $$M_{\kappa ,\mu }\left(z\right)=z^{\mu +\frac{1}{2}}\left(1+O\left({\mu }^{-1}\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function and $z$: complex variable

Referenced by:

§13.20(i)

Permalink:

http://dlmf.nist.gov/13.20.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.20(i), §13.20 and Ch.13

… ►

13.20.2 $$W_{\kappa ,\mu }\left(x\right)={\pi }^{-\frac{1}{2}}\mathrm{\Gamma }\left(\kappa +\mu \right){\left(\frac{1}{4}x\right)}^{\frac{1}{2}-\mu }\left(1+O\left({\mu }^{-1}\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter and $x$: real variable

Referenced by:

§13.20(i)

Permalink:

http://dlmf.nist.gov/13.20.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.20(i), §13.20 and Ch.13

… ►

13.20.4 $$M_{\kappa ,\mu }\left(x\right)=\sqrt{\frac{2\mu x}{X}}{\left(\frac{4{\mu }^{2}x}{2{\mu }^2-\kappa x+\mu X}\right)}^{\mu }{\left(\frac{2(\mu -\kappa )}{X+x-2\kappa }\right)}^{\kappa }{\mathrm{e}}^{\frac{1}{2}X-\mu }\left(1+O\left(\frac{1}{\mu }\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $x$: real variable and $X$

Permalink:

http://dlmf.nist.gov/13.20.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §13.20(ii), §13.20 and Ch.13

►

13.20.5 $$W_{\kappa ,\mu }\left(x\right)=\sqrt{\frac{x}{X}}{\left(\frac{2{\mu }^2-\kappa x+\mu X}{(\mu -\kappa )x}\right)}^{\mu }{\left(\frac{X+x-2\kappa }{2}\right)}^{\kappa }{\mathrm{e}}^{-\frac{1}{2}X-\kappa }\left(1+O\left(\frac{1}{\mu }\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $x$: real variable and $X$

Permalink:

http://dlmf.nist.gov/13.20.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §13.20(ii), §13.20 and Ch.13

… ►

13.20.11 $$W_{\kappa ,\mu }\left(x\right)={\left(\frac{1}{2}\mu \right)}^{-\frac{1}{4}}{\left(\frac{\kappa +\mu }{\mathrm{e}}\right)}^{\frac{1}{2}(\kappa +\mu )}\mathrm{\Phi }(\kappa ,\mu ,x)U(\mu -\kappa ,\zeta \sqrt{2\mu })\left(1+O\left({\mu }^{-1}\ln \mu \right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\ln z$: principal branch of logarithm function, $U(a,z)$: parabolic cylinder function, $x$: real variable and $\mathrm{\Phi }(\kappa ,\mu ,x)$: function

Referenced by:

§13.20(iv)

Permalink:

http://dlmf.nist.gov/13.20.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §13.20(iii), §13.20 and Ch.13

…

14: 13.21 Uniform Asymptotic Approximations for Large $\kappa$

… ►

13.21.1 $$M_{\kappa ,\mu }\left(x\right)=\sqrt{x}\mathrm{\Gamma }\left(2\mu +1\right){\kappa }^{-\mu }\left(J_{2\mu }\left(2\sqrt{x\kappa }\right)+\mathrm{env}{J}_{2\mu }\left(2\sqrt{x\kappa }\right)O\left({\kappa }^{-\frac{1}{2}}\right)\right),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{env}{J}_{\nu }\left(x\right)$: envelope of Bessel function $J_{\nu }\left(x\right)$ and $x$: real variable

Permalink:

http://dlmf.nist.gov/13.21.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.21(i), §13.21 and Ch.13

►

13.21.2 $$W_{\kappa ,\mu }\left(x\right)=\sqrt{x}\mathrm{\Gamma }\left(\kappa +\frac{1}{2}\right)\left(\sin \left(\kappa \pi -\mu \pi \right)J_{2\mu }\left(2\sqrt{x\kappa }\right)-\cos \left(\kappa \pi -\mu \pi \right)Y_{2\mu }\left(2\sqrt{x\kappa }\right)+\mathrm{env}{Y}_{2\mu }\left(2\sqrt{x\kappa }\right)O\left({\kappa }^{-\frac{1}{2}}\right)\right),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{env}{Y}_{\nu }\left(x\right)$: envelope of Bessel function $Y_{\nu }\left(x\right)$ , $\sin z$: sine function and $x$: real variable

Referenced by:

§13.21(i)

Permalink:

http://dlmf.nist.gov/13.21.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.21(i), §13.21 and Ch.13

►

13.21.3 $$W_{-\kappa ,\mu }\left(x{\mathrm{e}}^{-\pi \mathrm{i}}\right)=\frac{\pi \sqrt{x}}{\mathrm{\Gamma }\left(\kappa +\frac{1}{2}\right)}{\mathrm{e}}^{\mu \pi \mathrm{i}}\left(H_{2\mu }^{(1)}\left(2\sqrt{x\kappa }\right)+\mathrm{env}{Y}_{2\mu }\left(2\sqrt{x\kappa }\right)O\left({\kappa }^{-\frac{1}{2}}\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{env}{Y}_{\nu }\left(x\right)$: envelope of Bessel function $Y_{\nu }\left(x\right)$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $x$: real variable

Permalink:

http://dlmf.nist.gov/13.21.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.21(i), §13.21 and Ch.13

… ►

13.21.6 $$M_{-\kappa ,\mu }\left(4\kappa x\right)=\frac{2\mathrm{\Gamma }\left(2\mu +1\right)}{{\kappa }^{\mu -\frac{1}{2}}}{\left(\frac{x\zeta }{1+x}\right)}^{\frac{1}{4}}{I}_{2\mu }\left(4\kappa {\zeta }^{\frac{1}{2}}\right)\left(1+O\left({\kappa }^{-1}\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind and $x$: real variable

Referenced by:

§13.21(i)

Permalink:

http://dlmf.nist.gov/13.21.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §13.21(i), §13.21 and Ch.13

►

13.21.7 $$W_{-\kappa ,\mu }\left(4\kappa x\right)=\frac{\sqrt{8/\pi }{\mathrm{e}}^{\kappa }}{{\kappa }^{\kappa -\frac{1}{2}}}{\left(\frac{x\zeta }{1+x}\right)}^{\frac{1}{4}}{K}_{2\mu }\left(4\kappa {\zeta }^{\frac{1}{2}}\right)\left(1+O\left({\kappa }^{-1}\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind and $x$: real variable

Referenced by:

§13.21(i)

Permalink:

http://dlmf.nist.gov/13.21.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §13.21(i), §13.21 and Ch.13

…

15: 28.4 Fourier Series

… ►

28.4.21 $$A_{2s}^0(q)=\left(\frac{{(-1)}^{s}2}{{(s!)}^2}{\left(\frac{q}{4}\right)}^s+O\left(q^{s+2}\right)\right)A_0^0(q),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $!$: factorial (as in $n!$), $q=h^2$: parameter and $A_m(q)$: Fourier coefficient

A&S Ref:

20.2.29 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.4.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §28.4(vi), §28.4 and Ch.28

►

28.4.22 $$\begin{array}{c}{A}_{m+2s}^m(q)\\ B_{m+2s}^m(q)\end{array}\}=\left(\frac{{(-1)}^{s}m!}{s!(m+s)!}{\left(\frac{q}{4}\right)}^s+O\left(q^{s+1}\right)\right)\{\begin{array}{c}{A}_m^m(q),\\ B_m^m(q),\end{array}$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $!$: factorial (as in $n!$), $m$: integer, $q=h^2$: parameter, $A_m(q)$: Fourier coefficient and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.4.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §28.4(vi), §28.4 and Ch.28

►

28.4.23 $$\begin{array}{c}{A}_{m-2s}^m(q)\\ B_{m-2s}^m(q)\end{array}\}=\left(\frac{(m-s-1)!}{s!(m-1)!}{\left(\frac{q}{4}\right)}^s+O\left(q^{s+1}\right)\right)\{\begin{array}{c}{A}_m^m(q),\\ B_m^m(q).\end{array}$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $!$: factorial (as in $n!$), $m$: integer, $q=h^2$: parameter, $A_m(q)$: Fourier coefficient and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.4.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §28.4(vi), §28.4 and Ch.28

… ►

28.4.24 $$\frac{A_{2m}^{2n}(q)}{A_0^{2n}(q)}=\frac{{(-1)}^m}{{(m!)}^2}{\left(\frac{q}{4}\right)}^m\frac{\pi \left(1+O\left(m^{-1}\right)\right)}{w_{\text{II}}(\frac{1}{2}\pi ;a_{2n}\left(q\right),q)},$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $a_n\left(q\right)$: eigenvalues of Mathieu equation, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $m$: integer, $q=h^2$: parameter, $n$: integer, $w_{\text{II}}(z;a,q)$: fundamental solution and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.4.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §28.4(vii), §28.4 and Ch.28

►

28.4.25 $$\frac{A_{2m+1}^{2n+1}(q)}{A_1^{2n+1}(q)}=\frac{{(-1)}^{m+1}}{{\left({\left(\frac{1}{2}\right)}_{m+1}\right)}^2}{\left(\frac{q}{4}\right)}^{m+1}\frac{2\left(1+O\left(m^{-1}\right)\right)}{w_{\text{II}}^{\prime }(\frac{1}{2}\pi ;a_{2n+1}\left(q\right),q)},$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $a_n\left(q\right)$: eigenvalues of Mathieu equation, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $m$: integer, $q=h^2$: parameter, $n$: integer, $w(z)$: Mathieu’s equation solution and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.4.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §28.4(vii), §28.4 and Ch.28

…

16: 27.12 Asymptotic Formulas: Primes

… ►

27.12.5 $$\left|\pi \left(x\right)-\mathrm{li}\left(x\right)\right|=O\left(x\exp \left(-c{(\ln x)}^{1/2}\right)\right),$$ $x\to \mathrm{\infty }$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\exp z$: exponential function, $\mathrm{li}\left(x\right)$: logarithmic integral, $\ln z$: principal branch of logarithm function, $\pi \left(x\right)$: number of primes not exceeding a number and $x$: real number

Referenced by:

Erratum (V1.0.18) for Equation (27.12.5)

Permalink:

http://dlmf.nist.gov/27.12.E5

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.18):

Replaced $\sqrt{\ln x}$ with ${(\ln x)}^{1/2}$ to be consistent with other equations in this section.

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.12, §27.12 and Ch.27

… ►

27.12.6 $$\left|\pi \left(x\right)-\mathrm{li}\left(x\right)\right|=O\left(x\exp \left(-d{(\ln x)}^{3/5}{(\ln \ln x)}^{-1/5}\right)\right).$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\exp z$: exponential function, $\mathrm{li}\left(x\right)$: logarithmic integral, $\ln z$: principal branch of logarithm function, $\pi \left(x\right)$: number of primes not exceeding a number, $d$: positive integer and $x$: real number

Permalink:

http://dlmf.nist.gov/27.12.E6

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.12, §27.12 and Ch.27

… ►

27.12.8 $$\frac{\mathrm{li}\left(x\right)}{\varphi \left(m\right)}+O\left(x\exp \left(-\lambda (\alpha ){(\ln x)}^{1/2}\right)\right),$$ $m\le {(\ln x)}^{\alpha }$, $\alpha >0$,

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\varphi \left(n\right)$: Euler’s totient, $\exp z$: exponential function, $\mathrm{li}\left(x\right)$: logarithmic integral, $\ln z$: principal branch of logarithm function, $m$: positive integer and $x$: real number

Referenced by:

Erratum (V1.0.17) for Equation (27.12.8)

Permalink:

http://dlmf.nist.gov/27.12.E8

Encodings:

pMML, png, TeX

Errata (effective with 1.0.17):

Originally the first term was given incorrectly by $\frac{x}{\varphi \left(m\right)}$.

Reported 2017-12-04 by Gergő Nemes

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.12, §27.12 and Ch.27

…

17: 6.16 Mathematical Applications

… ►

6.16.4 $$R_n(x)=O\left(n^{-1}\right),$$ $n\to \mathrm{\infty }$,

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $x$: real variable, $n$: nonnegative integer and $R_n(x)$: remainder term

Permalink:

http://dlmf.nist.gov/6.16.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §6.16(i), §6.16 and Ch.6

… ►

6.16.5 $$\mathrm{li}\left(x\right)-\pi (x)=O\left(\sqrt{x}\ln x\right),$$ $x\to \mathrm{\infty }$,

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{li}\left(x\right)$: logarithmic integral, $\ln z$: principal branch of logarithm function, $x$: real variable and $\pi (x)$: number of primes $\le x$

A&S Ref:

5.1.50

Permalink:

http://dlmf.nist.gov/6.16.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §6.16(ii), §6.16 and Ch.6

…

18: 10.24 Functions of Imaginary Order

… ►

$${\tilde{J}}_{\nu }\left(x\right)=\sqrt{2/(\pi x)}\cos \left(x-\frac{1}{4}\pi \right)+O\left(x^{-\frac{3}{2}}\right),$$

►

$${\tilde{Y}}_{\nu }\left(x\right)=\sqrt{2/(\pi x)}\sin \left(x-\frac{1}{4}\pi \right)+O\left(x^{-\frac{3}{2}}\right).$$

… ►

10.24.7 $${\tilde{J}}_{\nu }\left(x\right)={\left(\frac{2\tanh \left(\frac{1}{2}\pi \nu \right)}{\pi \nu }\right)}^{\frac{1}{2}}\cos \left(\nu \ln \left(\frac{1}{2}x\right)-{\gamma }_{\nu }\right)+O\left(x^2\right),$$

ⓘ

Symbols:

${\tilde{J}}_{\nu }\left(x\right)$: Bessel function of imaginary order, $O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\tanh z$: hyperbolic tangent function, $\ln z$: principal branch of logarithm function, $x$: real variable, $\nu$: complex parameter and ${\gamma }_{\nu }$: parameter

Referenced by:

§10.24, §10.24

Permalink:

http://dlmf.nist.gov/10.24.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.24 and Ch.10

►

10.24.8 $${\tilde{Y}}_{\nu }\left(x\right)={\left(\frac{2\coth \left(\frac{1}{2}\pi \nu \right)}{\pi \nu }\right)}^{\frac{1}{2}}\sin \left(\nu \ln \left(\frac{1}{2}x\right)-{\gamma }_{\nu }\right)+O\left(x^2\right),$$ $\nu >0$,

ⓘ

Symbols:

${\tilde{Y}}_{\nu }\left(x\right)$: Bessel function of imaginary order, $O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\coth z$: hyperbolic cotangent function, $\ln z$: principal branch of logarithm function, $\sin z$: sine function, $x$: real variable, $\nu$: complex parameter and ${\gamma }_{\nu }$: parameter

Permalink:

http://dlmf.nist.gov/10.24.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §10.24 and Ch.10

… ►

10.24.9 $${\tilde{Y}}_0\left(x\right)=Y_0\left(x\right)=\frac{2}{\pi }\left(\ln \left(\frac{1}{2}x\right)+\gamma \right)+O\left(x^2\ln x\right),$$

ⓘ

Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, ${\tilde{Y}}_{\nu }\left(x\right)$: Bessel function of imaginary order, $O\left(x\right)$: order not exceeding, $\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function and $x$: real variable

Referenced by:

§10.24, §10.24

Permalink:

http://dlmf.nist.gov/10.24.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §10.24 and Ch.10

…

19: 10.45 Functions of Imaginary Order

… ►

$${\tilde{I}}_{\nu }\left(x\right)={(2\pi x)}^{-\frac{1}{2}}{\mathrm{e}}^x\left(1+O\left(x^{-1}\right)\right),$$

►

$${\tilde{K}}_{\nu }\left(x\right)={(\pi /(2x))}^{\frac{1}{2}}{\mathrm{e}}^{-x}\left(1+O\left(x^{-1}\right)\right).$$

… ►

10.45.6 $${\tilde{I}}_{\nu }\left(x\right)={\left(\frac{\sinh \left(\pi \nu \right)}{\pi \nu }\right)}^{\frac{1}{2}}\cos \left(\nu \ln \left(\frac{1}{2}x\right)-{\gamma }_{\nu }\right)+O\left(x^2\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sinh z$: hyperbolic sine function, ${\tilde{I}}_{\nu }\left(x\right)$: modified Bessel function of the first kind of imaginary order, $\ln z$: principal branch of logarithm function, $x$: real variable, $\nu$: complex parameter and ${\gamma }_{\nu }$: parameter

Referenced by:

§10.45

Permalink:

http://dlmf.nist.gov/10.45.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.45 and Ch.10

… ►

10.45.7 $${\tilde{K}}_{\nu }\left(x\right)=-{\left(\frac{\pi }{\nu \sinh \left(\pi \nu \right)}\right)}^{\frac{1}{2}}\sin \left(\nu \ln \left(\frac{1}{2}x\right)-{\gamma }_{\nu }\right)+O\left(x^2\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\sinh z$: hyperbolic sine function, ${\tilde{K}}_{\nu }\left(x\right)$: modified Bessel function fo the second kind of imaginary order, $\ln z$: principal branch of logarithm function, $\sin z$: sine function, $x$: real variable, $\nu$: complex parameter and ${\gamma }_{\nu }$: parameter

Referenced by:

§10.30(i), §10.45

Permalink:

http://dlmf.nist.gov/10.45.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.45 and Ch.10

… ►

10.45.8 $${\tilde{K}}_0\left(x\right)=K_0\left(x\right)=-\ln \left(\frac{1}{2}x\right)-\gamma +O\left(x^2\ln x\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\gamma$: Euler’s constant, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, ${\tilde{K}}_{\nu }\left(x\right)$: modified Bessel function fo the second kind of imaginary order, $\ln z$: principal branch of logarithm function and $x$: real variable

Referenced by:

§10.45

Permalink:

http://dlmf.nist.gov/10.45.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §10.45 and Ch.10

…

20: 13.8 Asymptotic Approximations for Large Parameters

… ►

13.8.1 $$M(a,b,z)=\sum _{s=0}^{n-1}\frac{{\left(a\right)}_s}{{\left(b\right)}_{s}s!}{z}^s+O\left({|b|}^{-n}\right).$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $n$: nonnegative integer, $s$: nonnegative integer and $z$: complex variable

Referenced by:

§13.9(ii)

Permalink:

http://dlmf.nist.gov/13.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.8(i), §13.8 and Ch.13

… ►

13.8.6 $$M(a,b,b)=\sqrt{\pi }{\left(\frac{b}{2}\right)}^{\frac{1}{2}a}\left(\frac{1}{\mathrm{\Gamma }\left(\frac{1}{2}(a+1)\right)}+\frac{(a+1)\sqrt{8/b}}{3\mathrm{\Gamma }\left(\frac{1}{2}a\right)}+O\left(\frac{1}{b}\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function and $\pi$: the ratio of the circumference of a circle to its diameter

Permalink:

http://dlmf.nist.gov/13.8.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §13.8(ii), §13.8 and Ch.13

… ►

13.8.7 $$U(a,b,b)=\sqrt{\pi }{\left(2b\right)}^{-\frac{1}{2}a}\left(\frac{1}{\mathrm{\Gamma }\left(\frac{1}{2}(a+1)\right)}-\frac{(a+1)\sqrt{8/b}}{3\mathrm{\Gamma }\left(\frac{1}{2}a\right)}+O\left(\frac{1}{b}\right)\right).$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function and $\pi$: the ratio of the circumference of a circle to its diameter

Permalink:

http://dlmf.nist.gov/13.8.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §13.8(ii), §13.8 and Ch.13

… ►

13.8.17 $$M(a,b,z)={\mathrm{e}}^{\nu z}\frac{{\mathrm{\Gamma }}^{\ast }(b)}{{\mathrm{\Gamma }}^{\ast }(a)}\left(1+\frac{(1-\nu )(1+6{\nu }^2{z}^2)}{12a}+O\left(\frac{1}{\min (a^2,b^2)}\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.8.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §13.8(iv), §13.8 and Ch.13

►

13.8.18 $$U(a,b+1,z)=z^{-b}{\mathrm{e}}^{(1-\nu )z}\frac{\mathrm{\Gamma }\left(b\right)}{\mathrm{\Gamma }\left(a\right)}\left(1+\frac{\nu z(1-\nu )(2-\nu z)}{2a}+O\left(\frac{1}{\min (a^2,b^2)}\right)\right),$$ $\mathrm{\Re }z>0$,

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Re }$: real part and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.8.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §13.8(iv), §13.8 and Ch.13

…