modified functions

SearchAdvancedHelp

(0.009 seconds)

31—40 of 174 matching pages

31: 11.4 Basic Properties

… ►

§11.4(i) Half-Integer Orders

… ►

§11.4(ii) Inequalities

… ►

§11.4(iii) Analytic Continuation

… ►

§11.4(v) Recurrence Relations and Derivatives

… ►

§11.4(vii) Zeros

…

32: 10.45 Functions of Imaginary Order

§10.45 Functions of Imaginary Order

… ►

10.45.2 ${\tilde{I}}_{\nu }\left(x\right)=\mathrm{\Re }\left(I_{\mathrm{i}\nu }\left(x\right)\right),$ ${\tilde{K}}_{\nu }\left(x\right)=K_{\mathrm{i}\nu }\left(x\right).$

ⓘ

Defines:

${\tilde{I}}_{\nu }\left(x\right)$: modified Bessel function of the first kind of imaginary order and ${\tilde{K}}_{\nu }\left(x\right)$: modified Bessel function fo the second kind of imaginary order

Symbols:

$\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part, $x$: real variable and $\nu$: complex parameter

Referenced by:

§10.30(i), §10.74(viii)

Permalink:

http://dlmf.nist.gov/10.45.E2

Encodings:

pMML, pMML, png, png, TeX, TeX

See also:

Annotations for §10.45 and Ch.10

… ►

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

… ► … ►

33: 11.15 Approximations

… ►

§11.15(i) Expansions in Chebyshev Series

…

34: 28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions

§28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions

… ►Also, with $I_n$ and $K_n$ denoting the modified Bessel functions (§10.25(ii)), and again with $s=0,1,2,\mathrm{\dots }$, … ►

28.24.11 $${\mathrm{Ko}}_{2m+2}(z,h)=\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}\frac{B_{2\mathrm{\ell }+2}^{2m+2}(h^2)}{B_{2s+2}^{2m+2}(h^2)}\left(I_{\mathrm{\ell }-s}\left(h{\mathrm{e}}^{-z}\right)K_{\mathrm{\ell }+s+2}\left(h{\mathrm{e}}^z\right)-I_{\mathrm{\ell }+s+2}\left(h{\mathrm{e}}^{-z}\right)K_{\mathrm{\ell }-s}\left(h{\mathrm{e}}^z\right)\right),$$

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, ${\mathrm{Ko}}_n(z,h)$: modified Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.24.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §28.24 and Ch.28

►

28.24.12 $${\mathrm{Ke}}_{2m+1}(z,h)=\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}\frac{B_{2\mathrm{\ell }+1}^{2m+1}(h^2)}{B_{2s+1}^{2m+1}(h^2)}\left(I_{\mathrm{\ell }-s}\left(h{\mathrm{e}}^{-z}\right)K_{\mathrm{\ell }+s+1}\left(h{\mathrm{e}}^z\right)-I_{\mathrm{\ell }+s+1}\left(h{\mathrm{e}}^{-z}\right)K_{\mathrm{\ell }-s}\left(h{\mathrm{e}}^z\right)\right),$$

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, ${\mathrm{Ke}}_n(z,h)$: modified Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.24.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §28.24 and Ch.28

… ►For further power series of Mathieu radial functions of integer order for small parameters and improved convergence rate see Larsen et al. (2009).

35: 10.34 Analytic Continuation

§10.34 Analytic Continuation

… ►

10.34.1 $$I_{\nu }\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={\mathrm{e}}^{m\nu \pi \mathrm{i}}{I}_{\nu }\left(z\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $m$: integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.30

Referenced by:

§10.30(ii), §10.34, §10.47(v)

Permalink:

http://dlmf.nist.gov/10.34.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.34 and Ch.10

►

10.34.2 $$K_{\nu }\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={\mathrm{e}}^{-m\nu \pi \mathrm{i}}{K}_{\nu }\left(z\right)-\pi \mathrm{i}\sin \left(m\nu \pi \right)\csc \left(\nu \pi \right)I_{\nu }\left(z\right).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\sin z$: sine function, $m$: integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.31

Referenced by:

§10.27, §10.34, §10.34

Permalink:

http://dlmf.nist.gov/10.34.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.34 and Ch.10

►

10.34.3 $$I_{\nu }\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)=(\mathrm{i}/\pi )\left(\pm {\mathrm{e}}^{m\nu \pi \mathrm{i}}{K}_{\nu }\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)\mp {\mathrm{e}}^{(m\mp 1)\nu \pi \mathrm{i}}{K}_{\nu }\left(z\right)\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $m$: integer, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.34, §10.40(i), §10.40(iii), §10.41(iv)

Permalink:

http://dlmf.nist.gov/10.34.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.34 and Ch.10

… ►

10.34.5 $$K_n\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={(-1)}^{mn}{K}_n\left(z\right)+{(-1)}^{n(m-1)-1}m\pi \mathrm{i}{I}_n\left(z\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $m$: integer, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/10.34.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.34 and Ch.10

…

36: 10.31 Power Series

§10.31 Power Series

… ►

10.31.1 $$K_n\left(z\right)=\frac{1}{2}{(\frac{1}{2}z)}^{-n}\sum _{k=0}^{n-1}\frac{(n-k-1)!}{k!}{(-\frac{1}{4}{z}^2)}^k+{(-1)}^{n+1}\ln \left(\frac{1}{2}z\right)I_n\left(z\right)+{(-1)}^n\frac{1}{2}{(\frac{1}{2}z)}^n\sum _{k=0}^{\mathrm{\infty }}\left(\psi \left(k+1\right)+\psi \left(n+k+1\right)\right)\frac{{(\frac{1}{4}{z}^2)}^k}{k!(n+k)!},$$

ⓘ

Symbols:

$\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\ln z$: principal branch of logarithm function, $n$: integer, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

9.6.11

Referenced by:

§10.30(i), §10.65(ii)

Permalink:

http://dlmf.nist.gov/10.31.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.31 and Ch.10

… ►

10.31.2 $$K_0\left(z\right)=-\left(\ln \left(\frac{1}{2}z\right)+\gamma \right)I_0\left(z\right)+\frac{\frac{1}{4}{z}^2}{{(1!)}^2}+(1+\frac{1}{2})\frac{{(\frac{1}{4}{z}^2)}^2}{{(2!)}^2}+(1+\frac{1}{2}+\frac{1}{3})\frac{{(\frac{1}{4}{z}^2)}^3}{{(3!)}^2}+\mathrm{\cdots }.$$

ⓘ

Symbols:

$\gamma$: Euler’s constant, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

9.6.13

Referenced by:

§10.45

Permalink:

http://dlmf.nist.gov/10.31.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.31 and Ch.10

… ►

10.31.3 $$I_{\nu }\left(z\right)I_{\mu }\left(z\right)={(\frac{1}{2}z)}^{\nu +\mu }\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(\nu +\mu +k+1\right)}_k{(\frac{1}{4}{z}^2)}^k}{k!\mathrm{\Gamma }\left(\nu +k+1\right)\mathrm{\Gamma }\left(\mu +k+1\right)}.$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.31

Permalink:

http://dlmf.nist.gov/10.31.E3

Encodings:

pMML, png, TeX

Correction (effective with 1.1.2):

The Pochhammer symbol now links to its definition.

See also:

Annotations for §10.31 and Ch.10

37: 28.35 Tables

§28.35 Tables

… ►

Kirkpatrick (1960) contains tables of the modified functions ${\mathrm{Ce}}_n(x,q)$, ${\mathrm{Se}}_{n+1}(x,q)$ for $n=0(1)5$, $q=1(1)20$, $x=0.1(.1)1$; 4D or 5D.

… ►

Zhang and Jin (1996, pp. 521–532) includes the eigenvalues $a_n\left(q\right)$, $b_{n+1}\left(q\right)$ for $n=0(1)4$, $q=0(1)50$; $n=0(1)20$ ($a$’s) or 19 ($b$’s), $q=1,3,5,10,15,25,50(50)200$. Fourier coefficients for ${\mathrm{ce}}_n(x,10)$, ${\mathrm{se}}_{n+1}(x,10)$, $n=0(1)7$. Mathieu functions ${\mathrm{ce}}_n(x,10)$, ${\mathrm{se}}_{n+1}(x,10)$, and their first $x$-derivatives for $n=0(1)4$, $x=0(5^{\circ }){90}^{\circ }$. Modified Mathieu functions ${\mathrm{Mc}}_n^{(j)}(x,\sqrt{10})$, ${\mathrm{Ms}}_{n+1}^{(j)}(x,\sqrt{10})$, and their first $x$-derivatives for $n=0(1)4$, $j=1,2$, $x=0(.2)4$. Precision is mostly 9S.

… ►

Blanch and Clemm (1969) includes eigenvalues $a_n\left(q\right)$, $b_n\left(q\right)$ for $q=\rho {\mathrm{e}}^{\mathrm{i}\varphi }$, $\rho =0(.5)25$, $\varphi =5^{\circ }(5^{\circ }){90}^{\circ }$, $n=0(1)15$; 4D. Also $a_n\left(q\right)$ and $b_n\left(q\right)$ for $q=\mathrm{i}\rho$, $\rho =0(.5)100$, $n=0(2)14$ and $n=2(2)16$, respectively; 8D. Double points for $n=0(1)15$; 8D. Graphs are included.

►

§28.35(iii) Zeros

…

38: 28.1 Special Notation

… ► ►►►

$m,n$	integers.
…
$\nu$	order of the Mathieu function or modified Mathieu function. (When $\nu$ is an integer it is often replaced by $n$.)
…

… ►and the modified Mathieu functions ► ►►

${\mathrm{Ce}}_{\nu }(z,q)$,	${\mathrm{Se}}_{\nu }(z,q)$,	${\mathrm{Fe}}_n(z,q)$,	${\mathrm{Ge}}_n(z,q)$,
…

►The functions ${\mathrm{Mc}}_n^{(j)}(z,h)$ and ${\mathrm{Ms}}_n^{(j)}(z,h)$ are also known as the radial Mathieu functions. … ►The radial functions ${\mathrm{Mc}}_n^{(j)}(z,h)$ and ${\mathrm{Ms}}_n^{(j)}(z,h)$ are denoted by ${\mathrm{Mc}}_n^{(j)}(z,q)$ and ${\mathrm{Ms}}_n^{(j)}(z,q)$, respectively.

39: 9.13 Generalized Airy Functions

… ►

9.13.2 $$w=z^{1/2}{𝒵}_p\left(\zeta \right),$$

ⓘ

Defines:

$w$: function (locally)

Symbols:

${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $z$: complex variable, $p$: variable and $\zeta$: variable

Source:

Swanson and Headley (1967, p. 1401)

Permalink:

http://dlmf.nist.gov/9.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §9.13(i), §9.13 and Ch.9

… ►and ${𝒵}_p$ is any linear combination of the modified Bessel functions $I_p$ and ${\mathrm{e}}^{p\pi \mathrm{i}}{K}_p$ (§10.25(ii)). ►Swanson and Headley (1967) define independent solutions $A_n\left(z\right)$ and $B_n\left(z\right)$ of (9.13.1) by … ►Properties of $A_n\left(z\right)$ and $B_n\left(z\right)$ follow from the corresponding properties of the modified Bessel functions. …

40: 13.18 Relations to Other Functions

… ►

§13.18(iii) Modified Bessel Functions

►When $\kappa =0$ the Whittaker functions can be expressed as modified Bessel functions. … ►

13.18.8 $$M_{0,\nu }\left(2z\right)=2^{2\nu +\frac{1}{2}}\mathrm{\Gamma }\left(1+\nu \right)\sqrt{z}{I}_{\nu }\left(z\right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: 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 $z$: complex variable

Referenced by:

§13.24(ii)

Permalink:

http://dlmf.nist.gov/13.18.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §13.18(iii), §13.18 and Ch.13

►

13.18.9 $$W_{0,\nu }\left(2z\right)=\sqrt{2z/\pi }{K}_{\nu }\left(z\right),$$

ⓘ

Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind and $z$: complex variable

Referenced by:

(18.34.2)

Permalink:

http://dlmf.nist.gov/13.18.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §13.18(iii), §13.18 and Ch.13

…