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11: 28.22 Connection Formulas

§28.22 Connection Formulas

… ►The joining factors in the above formulas are given by … ►

28.22.13 $${\mathrm{M}}_{\nu }^{(1)}(z,h)=\frac{{\mathrm{M}}_{\nu }^{(1)}(0,h)}{{\mathrm{me}}_{\nu }(0,h^2)}{\mathrm{Me}}_{\nu }(z,h^2).$$

ⓘ

Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, ${\mathrm{Me}}_{\nu }(z,q)$: modified Mathieu function, $h$: parameter, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.22.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §28.22(ii), §28.22 and Ch.28

►Here ${\mathrm{me}}_{\nu }(0,h^2)$ $(\ne 0)$ is given by (28.14.1) with $z=0$, and ${\mathrm{M}}_{\nu }^{(1)}(0,h)$ is given by (28.24.1) with $j=1$, $z=0$, and $n$ chosen so that $|c_{2n}^{\nu }(h^2)|=\max (|c_{2\mathrm{\ell }}^{\nu }(h^2)|)$, where the maximum is taken over all integers $\mathrm{\ell }$. … ►

12: 10.34 Analytic Continuation

§10.34 Analytic Continuation

… ►

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

… ►

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

ⓘ

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

… ►

$I_{\nu }\left(\overline{z}\right)=\overline{I_{\nu }\left(z\right)},$

►

$K_{\nu }\left(\overline{z}\right)=\overline{K_{\nu }\left(z\right)}.$

…

13: 10.33 Continued Fractions

§10.33 Continued Fractions

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

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

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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

…

14: 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 $s=0,1,2,\mathrm{\dots }$, … ►

28.24.10 $${\epsilon }_s{\mathrm{Ke}}_{2m}(z,h)=\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}\frac{A_{2\mathrm{\ell }}^{2m}(h^2)}{A_{2s}^{2m}(h^2)}\left(I_{\mathrm{\ell }-s}\left(h{\mathrm{e}}^{-z}\right)K_{\mathrm{\ell }+s}\left(h{\mathrm{e}}^z\right)+I_{\mathrm{\ell }+s}\left(h{\mathrm{e}}^{-z}\right)K_{\mathrm{\ell }-s}\left(h{\mathrm{e}}^z\right)\right),$$

ⓘ

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, ${\epsilon }_s$: constants and $A_m(q)$: Fourier coefficient

A&S Ref:

20.8.9

Permalink:

http://dlmf.nist.gov/28.24.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §28.24 and Ch.28

►

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

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

15: 28.35 Tables

… ►

Blanch and Clemm (1962) includes values of ${\mathrm{Mc}}_n^{(1)}(x,\sqrt{q})$ and $\mathrm{Mc}_n^{(1)}{}_{}{}^{\prime }(x,\sqrt{q})$ for $n=0(1)15$ with $q=0(.05)1$, $x=0(.02)1$. Also ${\mathrm{Ms}}_n^{(1)}(x,\sqrt{q})$ and $\mathrm{Ms}_n^{(1)}{}_{}{}^{\prime }(x,\sqrt{q})$ for $n=1(1)15$ with $q=0(.05)1$, $x=0(.02)1$. Precision is generally 7D.

►

Blanch and Clemm (1965) includes values of ${\mathrm{Mc}}_n^{(2)}(x,\sqrt{q})$, $\mathrm{Mc}_n^{(2)}{}_{}{}^{\prime }(x,\sqrt{q})$ for $n=0(1)7$, $x=0(.02)1$; $n=8(1)15$, $x=0(.01)1$. Also ${\mathrm{Ms}}_n^{(2)}(x,\sqrt{q})$, $\mathrm{Ms}_n^{(2)}{}_{}{}^{\prime }(x,\sqrt{q})$ for $n=1(1)7$, $x=0(.02)1$; $n=8(1)15$, $x=0(.01)1$. In all cases $q=0(.05)1$. Precision is generally 7D. Approximate formulas and graphs are also included.

… ►

§28.35(iii) Zeros

►

Blanch and Clemm (1965) includes the first and second zeros of ${\mathrm{Mc}}_n^{(2)}(x,\sqrt{q})$, $\mathrm{Mc}_n^{(2)}{}_{}{}^{\prime }(x,\sqrt{q})$ for $n=0,1$, and ${\mathrm{Ms}}_n^{(2)}(x,\sqrt{q})$, $\mathrm{Ms}_n^{(2)}{}_{}{}^{\prime }(x,\sqrt{q})$ for $n=1,2$, with $q=0(.05)1$; 7D.

… ►

Zhang and Jin (1996, pp. 533–535) includes the zeros (in degrees) of ${\mathrm{ce}}_n(x,10)$, ${\mathrm{se}}_n(x,10)$ for $n=1(1)10$, and the first 5 zeros of ${\mathrm{Mc}}_n^{(j)}(x,\sqrt{10})$, ${\mathrm{Ms}}_n^{(j)}(x,\sqrt{10})$ for $n=0$ or $1(1)8$, $j=1,2$. Precision is mostly 9S.

…

16: 10.25 Definitions

… ►

§10.25(i) Modified Bessel’s Equation

… ►Its solutions are called modified Bessel functions or Bessel functions of imaginary argument. ►

§10.25(ii) Standard Solutions

… ►

Branch Conventions

…

17: 10.36 Other Differential Equations

§10.36 Other Differential Equations

►The quantity ${\lambda }^2$ in (10.13.1)–(10.13.6) and (10.13.8) can be replaced by $-{\lambda }^2$ if at the same time the symbol $𝒞$ in the given solutions is replaced by $𝒵$. … ►

10.36.1 $$z^2(z^2+{\nu }^2)w^{\prime \prime }+z(z^2+3{\nu }^2)w^{\prime }-\left({(z^2+{\nu }^2)}^2+z^2-{\nu }^2\right)w=0,$$ $w={𝒵}_{\nu }^{\prime }\left(z\right)$,

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Symbols:

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

Permalink:

http://dlmf.nist.gov/10.36.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.36 and Ch.10

►

10.36.2 $$z^2{w}^{\prime \prime }+z(1\pm 2z)w^{\prime }+(\pm z-{\nu }^2)w=0,$$ $w={\mathrm{e}}^{\mp z}{𝒵}_{\nu }\left(z\right)$.

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, ${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.41

Permalink:

http://dlmf.nist.gov/10.36.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.36 and Ch.10

…

18: 10.31 Power Series

§10.31 Power Series

►For $I_{\nu }\left(z\right)$ see (10.25.2) and (10.27.1). When $\nu$ is not an integer the corresponding expansion for $K_{\nu }\left(z\right)$ is obtained from (10.25.2) and (10.27.4). … ►

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

…

19: 28.23 Expansions in Series of Bessel Functions

§28.23 Expansions in Series of Bessel Functions

… ►

28.23.2 $${\mathrm{me}}_{\nu }(0,h^2){\mathrm{M}}_{\nu }^{(j)}(z,h)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{(-1)}^n{c}_{2n}^{\nu }(h^2){𝒞}_{\nu +2n}^{(j)}(2h\cosh z),$$

ⓘ

Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\cosh z$: hyperbolic cosine function, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $n$: integer, $j$: integer, $z$: complex variable, $\nu$: complex parameter, $c_{2m}(q)$: coefficients and ${𝒞}_{\mu }^{(j)}$: cylinder functions

Permalink:

http://dlmf.nist.gov/28.23.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

… ►

28.23.6 $${\mathrm{Mc}}_{2m}^{(j)}(z,h)={(-1)}^m{\left({\mathrm{ce}}_{2m}(0,h^2)\right)}^{-1}\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{\ell }}{A}_{2\mathrm{\ell }}^{2m}(h^2){𝒞}_{2\mathrm{\ell }}^{(j)}(2h\cosh z),$$

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cosh z$: hyperbolic cosine function, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $A_m(q)$: Fourier coefficient

A&S Ref:

20.6.12

Referenced by:

§28.23

Permalink:

http://dlmf.nist.gov/28.23.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

►

28.23.7 $${\mathrm{Mc}}_{2m}^{(j)}(z,h)={(-1)}^m{\left({\mathrm{ce}}_{2m}(\frac{1}{2}\pi ,h^2)\right)}^{-1}\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{A}_{2\mathrm{\ell }}^{2m}(h^2){𝒞}_{2\mathrm{\ell }}^{(j)}(2h\sinh z),$$

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sinh z$: hyperbolic sine function, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.23.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

… ►

20: 10.37 Inequalities; Monotonicity

§10.37 Inequalities; Monotonicity

►If $\nu$ $(\ge 0)$ is fixed, then throughout the interval , $I_{\nu }\left(x\right)$ is positive and increasing, and $K_{\nu }\left(x\right)$ is positive and decreasing. ►If $x$ $(>0)$ is fixed, then throughout the interval , $I_{\nu }\left(x\right)$ is decreasing, and $K_{\nu }\left(x\right)$ is increasing. … ►

10.37.1

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Symbols:

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

Referenced by:

§10.37, §10.37, Erratum (V1.0.17) for Section 10.37

Permalink:

http://dlmf.nist.gov/10.37.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.37 and Ch.10

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