modified Mathieu functions

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1: 28.20 Definitions and Basic Properties

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§28.20(ii) Solutions ${\mathrm{Ce}}_{\nu }$, ${\mathrm{Se}}_{\nu }$, ${\mathrm{Me}}_{\nu }$, ${\mathrm{Fe}}_n$, ${\mathrm{Ge}}_n$

… ►For other values of $z$, $h$, and $\nu$ the functions ${\mathrm{M}}_{\nu }^{(j)}(z,h)$, $j=1,2,3,4$, are determined by analytic continuation. … ►

§28.20(iv) Radial Mathieu Functions ${\mathrm{Mc}}_n^{(j)}$, ${\mathrm{Ms}}_n^{(j)}$

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§28.20(vi) Wronskians

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§28.20(vii) Shift of Variable

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2: 28.27 Addition Theorems

§28.27 Addition Theorems

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3: 28.22 Connection Formulas

§28.22 Connection Formulas

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28.22.4 $${\mathrm{Ms}}_m^{(2)}(z,h)=\sqrt{\frac{2}{\pi }}\frac{1}{g_{o,m}(h){\mathrm{se}}_m^{\prime }(0,h^2)}\left(-f_{o,m}(h){\mathrm{Se}}_m(z,h^2)-\frac{2}{\pi S_m(h^2)}{\mathrm{Ge}}_m(z,h^2)\right).$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{Ge}}_n(z,q)$: modified Mathieu function, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, ${\mathrm{Se}}_{\nu }(z,q)$: modified Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable, $g_{e,n}(h)$, $g_{o,n}(h)$: joining factors, $f_{e,n}(h)$, $f_{o,n}(h)$: joining factors and $S_n(q)$: factor

Permalink:

http://dlmf.nist.gov/28.22.E4

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

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

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

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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

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

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4: 28.23 Expansions in Series of Bessel Functions

§28.23 Expansions in Series of Bessel Functions

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

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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

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Annotations for §28.23 and Ch.28

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

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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:

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See also:

Annotations for §28.23 and Ch.28

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28.23.8 $${\mathrm{Mc}}_{2m+1}^{(j)}(z,h)={(-1)}^m{\left({\mathrm{ce}}_{2m+1}(0,h^2)\right)}^{-1}\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{\ell }}{A}_{2\mathrm{\ell }+1}^{2m+1}(h^2){𝒞}_{2\mathrm{\ell }+1}^{(j)}(2h\cosh z),$$

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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

Permalink:

http://dlmf.nist.gov/28.23.E8

Encodings:

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See also:

Annotations for §28.23 and Ch.28

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5: 28.1 Special Notation

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

${\mathrm{Ce}}_{\nu }(z,q)$,	${\mathrm{Se}}_{\nu }(z,q)$,	${\mathrm{Fe}}_n(z,q)$,	${\mathrm{Ge}}_n(z,q)$,
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►The functions ${\mathrm{Mc}}_n^{(j)}(z,h)$ and ${\mathrm{Ms}}_n^{(j)}(z,h)$ are also known as the radial Mathieu functions. … ►

$f_{o,n}(h).$

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6: 28.35 Tables

§28.35 Tables

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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.

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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.

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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.

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§28.35(iii) Zeros

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7: 28.34 Methods of Computation

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§28.34(iv) Modified Mathieu Functions

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8: 28.33 Physical Applications

§28.33 Physical Applications

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28.33.2 $$V_n(\xi ,\eta )=\left(c_n{\mathrm{M}}_n^{(1)}(\xi ,\sqrt{q})+d_n{\mathrm{M}}_n^{(2)}(\xi ,\sqrt{q})\right){\mathrm{me}}_n(\eta ,q),$$

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $q=h^2$: parameter, $n$: integer, $V(\xi ,\eta )$: solution, $d_n$: constants, $\eta$: variable, $\xi$: variable and $c_n$: constants

Permalink:

http://dlmf.nist.gov/28.33.E2

Encodings:

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See also:

Annotations for §28.33(ii), §28.33 and Ch.28

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28.33.3 $${\mathrm{M}}_n^{(1)}({\xi }_0,\sqrt{q}){\mathrm{M}}_n^{(2)}({\xi }_1,\sqrt{q})-{\mathrm{M}}_n^{(1)}({\xi }_1,\sqrt{q}){\mathrm{M}}_n^{(2)}({\xi }_0,\sqrt{q})=0.$$

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Symbols:

${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $q=h^2$: parameter, $n$: integer and $\xi$: variable

Referenced by:

§28.33(ii)

Permalink:

http://dlmf.nist.gov/28.33.E3

Encodings:

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See also:

Annotations for §28.33(ii), §28.33 and Ch.28

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Torres-Vega et al. (1998) for Mathieu functions in phase space.

9: 28.28 Integrals, Integral Representations, and Integral Equations

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§28.28(i) Equations with Elementary Kernels

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28.28.11 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{2\mathrm{i}h\cosh z\cosh t}{\mathrm{Ce}}_{\nu }(t,h^2)dt=\frac{1}{2}\pi \mathrm{i}{\mathrm{e}}^{\mathrm{i}\nu \pi }{\mathrm{ce}}_{\nu }(0,h^2){\mathrm{M}}_{\nu }^{(3)}(z,h),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\int$: integral, ${\mathrm{Ce}}_{\nu }(z,q)$: modified Mathieu function, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§28.28(i)

Permalink:

http://dlmf.nist.gov/28.28.E11

Encodings:

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Annotations for §28.28(i), §28.28 and Ch.28

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28.28.15 $${\int }_0^{\mathrm{\infty }}\cos \left(2h\cos y\cosh t\right){\mathrm{Ce}}_{2n}(t,h^2)dt={(-1)}^{n+1}\frac{1}{2}\pi {\mathrm{Mc}}_{2n}^{(2)}(0,h){\mathrm{ce}}_{2n}(y,h^2),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\int$: integral, ${\mathrm{Ce}}_{\nu }(z,q)$: modified Mathieu function, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $h$: parameter, $n$: integer and $y$: real variable

Permalink:

http://dlmf.nist.gov/28.28.E15

Encodings:

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Annotations for §28.28(i), §28.28 and Ch.28

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§28.28(iv) Integrals of Products of Mathieu Functions of Integer Order

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§28.28(v) Compendia

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10: 28.21 Graphics

§28.21 Graphics

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Figure 28.21.6: ${\mathrm{Ms}}_1^{(2)}(x,h)$ for $0.2\le h\le 2$, $0\le x\le 2$. Magnify 3D Help