28 Mathieu Functions and Hill’s Equation Modified Mathieu Functions 28.19 Expansions in Series of $\operatorname{me}_{\nu+2n}$ Functions 28.21 Graphics

§28.20 Definitions and Basic Properties

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Permalink:: http://dlmf.nist.gov/28.20
See also:: Annotations for Ch.28

§28.20(i) Modified Mathieu’s Equation
§28.20(ii) Solutions $\operatorname{Ce}_{\nu}$ , $\operatorname{Se}_{\nu}$ , $\operatorname{Me}_{\nu}$ , $\operatorname{Fe}_{n}$ , $\operatorname{Ge}_{n}$
§28.20(iii) Solutions ${\operatorname{M}^{(j)}_{\nu}}$
§28.20(iv) Radial Mathieu Functions ${\operatorname{Mc}^{(j)}_{n}}$ , ${\operatorname{Ms}^{(j)}_{n}}$
§28.20(v) Solutions $\operatorname{Ie}_{n}$ , $\operatorname{Io}_{n}$ , $\operatorname{Ke}_{n}$ , $\operatorname{Ko}_{n}$
§28.20(vi) Wronskians
§28.20(vii) Shift of Variable

§28.20(i) Modified Mathieu’s Equation

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Keywords:: algebraic form, differential equation, modified Mathieu functions, modified Mathieu’s equation
Permalink:: http://dlmf.nist.gov/28.20.i
See also:: Annotations for §28.20 and Ch.28

When is replaced by $\pm\mathrm{i}z$ , (28.2.1) becomes the modified Mathieu’s equation:

28.20.1 $w^{\prime\prime}-\left(a-2q\cosh\left(2z\right)\right)w=0,$

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Symbols:: $\cosh\NVar{z}$ : hyperbolic cosine function, $q=h^{2}$ : parameter, : complex variable, : parameter and : Mathieu’s equation solution
A&S Ref:: 20.1.2 (in slightly different form) 20.8.6
Referenced by:: §28.20(iii), §28.32(i), 1st item, 2nd item, item (b), §28.8(iv), §28.8(iv)
Permalink:: http://dlmf.nist.gov/28.20.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §28.20(i), §28.20 and Ch.28

with its algebraic form

28.20.2 ${(\zeta^{2}-1)w^{\prime\prime}+\zeta w^{\prime}+\left(4q\zeta^{2}-2q-a\right)w% =0},$ $\zeta=\cosh z$ .

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Symbols:: $\cosh\NVar{z}$ : hyperbolic cosine function, $q=h^{2}$ : parameter, : complex variable, : parameter and : Mathieu’s equation solution
Referenced by:: §28.20(iii)
Permalink:: http://dlmf.nist.gov/28.20.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §28.20(i), §28.20 and Ch.28

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

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Keywords:: Mathieu functions, connection formulas, definitions, modified Mathieu functions, relation to Mathieu functions, relations to other functions
Notes:: See Meixner and Schäfke (1954, §§2.29 and 2.73).
Permalink:: http://dlmf.nist.gov/28.20.ii
See also:: Annotations for §28.20 and Ch.28

28.20.3 $\operatorname{Ce}_{\nu}\left(z,q\right)=\operatorname{ce}_{\nu}\left(\pm% \mathrm{i}z,q\right),$ $\nu\neq-1,-2,\dots$ ,

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Defines:: $\operatorname{Ce}_{\NVar{\nu}}\left(\NVar{z},\NVar{q}\right)$ : modified Mathieu function
Symbols:: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\mathrm{i}$ : imaginary unit, $q=h^{2}$ : parameter, : complex variable and $\nu$ : complex parameter
A&S Ref:: 20.6.1 (in slightly different form) 20.6.2 (in slightly different form)
Permalink:: http://dlmf.nist.gov/28.20.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §28.20(ii), §28.20 and Ch.28

28.20.4 $\operatorname{Se}_{\nu}\left(z,q\right)=\mp\mathrm{i}\operatorname{se}_{\nu}% \left(\pm\mathrm{i}z,q\right),$ $\nu\neq 0,-1,\dots$ ,

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Defines:: $\operatorname{Se}_{\NVar{\nu}}\left(\NVar{z},\NVar{q}\right)$ : modified Mathieu function
Symbols:: $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\mathrm{i}$ : imaginary unit, $q=h^{2}$ : parameter, : complex variable and $\nu$ : complex parameter
A&S Ref:: 20.6.1 (in slightly different form) 20.6.2 (in slightly different form)
Permalink:: http://dlmf.nist.gov/28.20.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §28.20(ii), §28.20 and Ch.28

28.20.5 $\operatorname{Me}_{\nu}\left(z,q\right)=\operatorname{me}_{\nu}\left(-\mathrm{% i}z,q\right),$

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Defines:: $\operatorname{Me}_{\NVar{\nu}}\left(\NVar{z},\NVar{q}\right)$ : modified Mathieu function
Symbols:: $\operatorname{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\mathrm{i}$ : imaginary unit, $q=h^{2}$ : parameter, : complex variable and $\nu$ : complex parameter
A&S Ref:: 20.6.1 (in slightly different form) 20.6.2 (in slightly different form)
Permalink:: http://dlmf.nist.gov/28.20.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §28.20(ii), §28.20 and Ch.28

28.20.6 $\operatorname{Fe}_{n}\left(z,q\right)=\mp\mathrm{i}\operatorname{fe}_{n}\left(% \pm\mathrm{i}z,q\right),$ $n=0,1,\dots$ ,

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Defines:: $\operatorname{Fe}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : modified Mathieu function
Symbols:: $\operatorname{fe}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : second solution, Mathieu’s equation, $\mathrm{i}$ : imaginary unit, $q=h^{2}$ : parameter, : integer and : complex variable
A&S Ref:: 20.6.1 (in slightly different form) 20.6.2 (in slightly different form)
Permalink:: http://dlmf.nist.gov/28.20.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §28.20(ii), §28.20 and Ch.28

28.20.7 $\operatorname{Ge}_{n}\left(z,q\right)=\operatorname{ge}_{n}\left(\pm\mathrm{i}% z,q\right),$ $n=1,2,\dots$ .

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Defines:: $\operatorname{Ge}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : modified Mathieu function
Symbols:: $\operatorname{ge}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : second solution, Mathieu’s equation, $\mathrm{i}$ : imaginary unit, $q=h^{2}$ : parameter, : integer and : complex variable
A&S Ref:: 20.6.1 (in slightly different form) 20.6.2 (in slightly different form)
Permalink:: http://dlmf.nist.gov/28.20.E7
Encodings:: TeX, pMML, png
See also:: Annotations for §28.20(ii), §28.20 and Ch.28

§28.20(iii) Solutions ${\operatorname{M}^{(j)}_{\nu}}$

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Defines:: ${\operatorname{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function
Keywords:: analytic continuation, asymptotic approximations, for large $\Re z$ , modified Mathieu functions
Notes:: See Meixner and Schäfke (1954, §2.41).
Referenced by:: §28.8(iv)
Permalink:: http://dlmf.nist.gov/28.20.iii
See also:: Annotations for §28.20 and Ch.28

Assume first that $\nu$ is real, is positive, and $a=\lambda_{\nu}\left(q\right)$ ; see §28.12(i). Write

28.20.8 $h=\sqrt{q}\;(>0).$

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Symbols:: $q=h^{2}$ : parameter and : parameter
Permalink:: http://dlmf.nist.gov/28.20.E8
Encodings:: TeX, pMML, png
See also:: Annotations for §28.20(iii), §28.20 and Ch.28

Then from §2.7(ii) it is seen that equation (28.20.2) has independent and unique solutions that are asymptotic to $\zeta^{\ifrac{1}{2}}e^{\pm 2\mathrm{i}h\zeta}$ as $\zeta\to\infty$ in the respective sectors $|\operatorname{ph}\left(\mp\mathrm{i}\zeta\right)|\leq\tfrac{3}{2}\pi-\delta$ , $\delta$ being an arbitrary small positive constant. It follows that (28.20.1) has independent and unique solutions ${\operatorname{M}^{(3)}_{\nu}}\left(z,h\right)$ , ${\operatorname{M}^{(4)}_{\nu}}\left(z,h\right)$ such that

28.20.9 ${\operatorname{M}^{(3)}_{\nu}}\left(z,h\right)={H^{(1)}_{\nu}}\left(2h\cosh z% \right)\left(1+O\left(\operatorname{sech}z\right)\right),$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $\cosh\NVar{z}$ : hyperbolic cosine function, $\operatorname{sech}\NVar{z}$ : hyperbolic secant function, ${\operatorname{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function, : parameter, : complex variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/28.20.E9
Encodings:: TeX, pMML, png
See also:: Annotations for §28.20(iii), §28.20 and Ch.28

as $\Re z\to+\infty$ with $-\pi+\delta\leq\Im z\leq 2\pi-\delta$ , and

28.20.10 ${\operatorname{M}^{(4)}_{\nu}}\left(z,h\right)={H^{(2)}_{\nu}}\left(2h\cosh z% \right)\left(1+O\left(\operatorname{sech}z\right)\right),$

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $\cosh\NVar{z}$ : hyperbolic cosine function, $\operatorname{sech}\NVar{z}$ : hyperbolic secant function, ${\operatorname{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function, : parameter, : complex variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/28.20.E10
Encodings:: TeX, pMML, png
See also:: Annotations for §28.20(iii), §28.20 and Ch.28

as $\Re z\to+\infty$ with $-2\pi+\delta\leq\Im z\leq\pi-\delta$ . See §10.2(ii) for the notation. In addition, there are unique solutions ${\operatorname{M}^{(1)}_{\nu}}\left(z,h\right)$ , ${\operatorname{M}^{(2)}_{\nu}}\left(z,h\right)$ that are real when is real and have the properties

28.20.11 ${\operatorname{M}^{(1)}_{\nu}}\left(z,h\right)=J_{\nu}\left(2h\cosh z\right)+e% ^{|\Im\left(2h\cosh z\right)|}O\left(\left(\operatorname{sech}z\right)^{3/2}% \right),$

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $O\left(\NVar{x}\right)$ : order not exceeding, $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : hyperbolic cosine function, $\operatorname{sech}\NVar{z}$ : hyperbolic secant function, $\Im$ : imaginary part, ${\operatorname{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function, : parameter, : complex variable and $\nu$ : complex parameter
Referenced by:: §28.33(ii)
Permalink:: http://dlmf.nist.gov/28.20.E11
Encodings:: TeX, pMML, png
See also:: Annotations for §28.20(iii), §28.20 and Ch.28

28.20.12 ${\operatorname{M}^{(2)}_{\nu}}\left(z,h\right)=Y_{\nu}\left(2h\cosh z\right)+e% ^{|\Im\left(2h\cosh z\right)|}O\left((\operatorname{sech}z)^{3/2}\right),$

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Symbols:: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $O\left(\NVar{x}\right)$ : order not exceeding, $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : hyperbolic cosine function, $\operatorname{sech}\NVar{z}$ : hyperbolic secant function, $\Im$ : imaginary part, ${\operatorname{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function, : parameter, : complex variable and $\nu$ : complex parameter
Referenced by:: §28.33(ii)
Permalink:: http://dlmf.nist.gov/28.20.E12
Encodings:: TeX, pMML, png
See also:: Annotations for §28.20(iii), §28.20 and Ch.28

as $\Re z\to+\infty$ with $|\Im z|\leq\pi-\delta$ .

For other values of , , and $\nu$ the functions ${\operatorname{M}^{(j)}_{\nu}}\left(z,h\right)$ , , are determined by analytic continuation. Furthermore,

28.20.13 ${\operatorname{M}^{(3)}_{\nu}}\left(z,h\right)={\operatorname{M}^{(1)}_{\nu}}% \left(z,h\right)+\mathrm{i}{\operatorname{M}^{(2)}_{\nu}}\left(z,h\right),$

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Symbols:: $\mathrm{i}$ : imaginary unit, ${\operatorname{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function, : parameter, : complex variable and $\nu$ : complex parameter
A&S Ref:: 20.6.16 (only for integer $\nu$ )
Referenced by:: §28.22(ii)
Permalink:: http://dlmf.nist.gov/28.20.E13
Encodings:: TeX, pMML, png
See also:: Annotations for §28.20(iii), §28.20 and Ch.28

28.20.14 ${\operatorname{M}^{(4)}_{\nu}}\left(z,h\right)={\operatorname{M}^{(1)}_{\nu}}% \left(z,h\right)-\mathrm{i}{\operatorname{M}^{(2)}_{\nu}}\left(z,h\right).$

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Symbols:: $\mathrm{i}$ : imaginary unit, ${\operatorname{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function, : parameter, : complex variable and $\nu$ : complex parameter
Referenced by:: §28.22(ii)
Permalink:: http://dlmf.nist.gov/28.20.E14
Encodings:: TeX, pMML, png
See also:: Annotations for §28.20(iii), §28.20 and Ch.28

§28.20(iv) Radial Mathieu Functions ${\operatorname{Mc}^{(j)}_{n}}$ , ${\operatorname{Ms}^{(j)}_{n}}$

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Keywords:: definitions, radial Mathieu functions, relation to modified Mathieu functions
Permalink:: http://dlmf.nist.gov/28.20.iv
See also:: Annotations for §28.20 and Ch.28

For ,

28.20.15 ${\operatorname{Mc}^{(j)}_{n}}\left(z,h\right)={\operatorname{M}^{(j)}_{n}}% \left(z,h\right),$ $n=0,1,\dots$ ,

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Defines:: ${\operatorname{Mc}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function
Symbols:: ${\operatorname{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function, : parameter, : integer, : integer and : complex variable
Referenced by:: §28.20(vii)
Permalink:: http://dlmf.nist.gov/28.20.E15
Encodings:: TeX, pMML, png
See also:: Annotations for §28.20(iv), §28.20 and Ch.28

28.20.16 ${\operatorname{Ms}^{(j)}_{n}}\left(z,h\right)=(-1)^{n}{\operatorname{M}^{(j)}_% {-n}}\left(z,h\right),$ $n=1,2,\dots$ .

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Defines:: ${\operatorname{Ms}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function
Symbols:: ${\operatorname{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function, : parameter, : integer, : integer and : complex variable
Referenced by:: §28.20(vii)
Permalink:: http://dlmf.nist.gov/28.20.E16
Encodings:: TeX, pMML, png
See also:: Annotations for §28.20(iv), §28.20 and Ch.28

§28.20(v) Solutions $\operatorname{Ie}_{n}$ , $\operatorname{Io}_{n}$ , $\operatorname{Ke}_{n}$ , $\operatorname{Ko}_{n}$

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Permalink:: http://dlmf.nist.gov/28.20.v
See also:: Annotations for §28.20 and Ch.28

28.20.17 $\operatorname{Ie}_{n}\left(z,h\right)={\mathrm{i}}^{-n}{\operatorname{Mc}^{(1)% }_{n}}\left(z,\mathrm{i}h\right),$

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Defines:: $\operatorname{Ie}_{\NVar{n}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function
Symbols:: $\mathrm{i}$ : imaginary unit, ${\operatorname{Mc}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function, : parameter, : integer and : complex variable
A&S Ref:: 20.8.10 (in slightly different form)
Permalink:: http://dlmf.nist.gov/28.20.E17
Encodings:: TeX, pMML, png
See also:: Annotations for §28.20(v), §28.20 and Ch.28

28.20.18 $\operatorname{Io}_{n}\left(z,h\right)={\mathrm{i}}^{-n}{\operatorname{Ms}^{(1)% }_{n}}\left(z,\mathrm{i}h\right),$

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Defines:: $\operatorname{Io}_{\NVar{n}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function
Symbols:: $\mathrm{i}$ : imaginary unit, ${\operatorname{Ms}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function, : parameter, : integer and : complex variable
A&S Ref:: 20.8.10 (in slightly different form)
Permalink:: http://dlmf.nist.gov/28.20.E18
Encodings:: TeX, pMML, png
See also:: Annotations for §28.20(v), §28.20 and Ch.28

28.20.19

$\operatorname{Ke}_{2m}\left(z,h\right)=(-1)^{m}\tfrac{1}{2}\pi\mathrm{i}{% \operatorname{Mc}^{(3)}_{2m}}\left(z,\mathrm{i}h\right),$

$\operatorname{Ke}_{2m+1}\left(z,h\right)=(-1)^{m+1}\tfrac{1}{2}\pi{% \operatorname{Mc}^{(3)}_{2m+1}}\left(z,\mathrm{i}h\right),$

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Defines:: $\operatorname{Ke}_{\NVar{n}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function
Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit, ${\operatorname{Mc}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function, : integer, : parameter, : integer and : complex variable
A&S Ref:: 20.8.11 (in slightly different form)
Permalink:: http://dlmf.nist.gov/28.20.E19
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §28.20(v), §28.20 and Ch.28

28.20.20

$\operatorname{Ko}_{2m}\left(z,h\right)=(-1)^{m}\tfrac{1}{2}\pi\mathrm{i}{% \operatorname{Ms}^{(3)}_{2m}}\left(z,\mathrm{i}h\right),$

$\operatorname{Ko}_{2m+1}\left(z,h\right)=(-1)^{m+1}\tfrac{1}{2}\pi{% \operatorname{Ms}^{(3)}_{2m+1}}\left(z,\mathrm{i}h\right).$

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Defines:: $\operatorname{Ko}_{\NVar{n}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function
Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit, ${\operatorname{Ms}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function, : integer, : parameter, : integer and : complex variable
Permalink:: http://dlmf.nist.gov/28.20.E20
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §28.20(v), §28.20 and Ch.28

§28.20(vi) Wronskians

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Keywords:: Wronskians, modified Mathieu functions
Notes:: See Meixner and Schäfke (1954, pp. 170–171).
Permalink:: http://dlmf.nist.gov/28.20.vi
See also:: Annotations for §28.20 and Ch.28

28.20.21

$\mathscr{W}\left\{{\operatorname{M}^{(1)}_{\nu}},{\operatorname{M}^{(2)}_{\nu}% }\right\}=-\mathscr{W}\left\{{\operatorname{M}^{(2)}_{\nu}},{\operatorname{M}^% {(3)}_{\nu}}\right\}={-\mathscr{W}\left\{{\operatorname{M}^{(2)}_{\nu}},{% \operatorname{M}^{(4)}_{\nu}}\right\}=\ifrac{2}{\pi}},$

$\mathscr{W}\left\{{\operatorname{M}^{(1)}_{\nu}},{\operatorname{M}^{(3)}_{\nu}% }\right\}=-\mathscr{W}\left\{{\operatorname{M}^{(1)}_{\nu}},{\operatorname{M}^% {(4)}_{\nu}}\right\}={-\tfrac{1}{2}\mathscr{W}\left\{{\operatorname{M}^{(3)}_{% \nu}},{\operatorname{M}^{(4)}_{\nu}}\right\}=\ifrac{2\mathrm{i}}{\pi}}.$

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Symbols:: $\mathscr{W}$ : Wronskian, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit, ${\operatorname{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/28.20.E21
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §28.20(vi), §28.20 and Ch.28

§28.20(vii) Shift of Variable

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Keywords:: modified Mathieu functions, radial Mathieu functions, shift of variable
Notes:: See Meixner and Schäfke (1954, §2.75 and p. 170).
Permalink:: http://dlmf.nist.gov/28.20.vii
See also:: Annotations for §28.20 and Ch.28

28.20.22 ${\operatorname{M}^{(j)}_{\nu}}\left(z\pm\tfrac{1}{2}\pi\mathrm{i},h\right)={% \operatorname{M}^{(j)}_{\nu}}\left(z,\pm\mathrm{i}h\right),$ $\nu\notin\mathbb{Z}$ .

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit, $\mathbb{Z}$ : set of all integers, ${\operatorname{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function, $\notin$ : not an element of, : parameter, : integer, : complex variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/28.20.E22
Encodings:: TeX, pMML, png
See also:: Annotations for §28.20(vii), §28.20 and Ch.28

For $n=0,1,2,\dots$ ,

28.20.23

${\operatorname{Mc}^{(j)}_{2n}}\left(z\pm\tfrac{1}{2}\pi\mathrm{i},h\right)={% \operatorname{Mc}^{(j)}_{2n}}\left(z,\pm\mathrm{i}h\right),$

${\operatorname{Ms}^{(j)}_{2n+1}}\left(z\pm\tfrac{1}{2}\pi\mathrm{i},h\right)={% \operatorname{Mc}^{(j)}_{2n+1}}\left(z,\pm\mathrm{i}h\right),$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit, ${\operatorname{Mc}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function, ${\operatorname{Ms}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function, : parameter, : integer, : integer and : complex variable
A&S Ref:: 20.8.7 (in slightly different form)
Permalink:: http://dlmf.nist.gov/28.20.E23
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §28.20(vii), §28.20 and Ch.28

28.20.24

${\operatorname{Mc}^{(j)}_{2n+1}}\left(z\pm\tfrac{1}{2}\pi\mathrm{i},h\right)={% \operatorname{Ms}^{(j)}_{2n+1}}\left(z,\pm\mathrm{i}h\right),$

${\operatorname{Ms}^{(j)}_{2n+2}}\left(z\pm\tfrac{1}{2}\pi\mathrm{i},h\right)={% \operatorname{Ms}^{(j)}_{2n+2}}\left(z,\pm\mathrm{i}h\right).$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit, ${\operatorname{Mc}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function, ${\operatorname{Ms}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function, : parameter, : integer, : integer and : complex variable
A&S Ref:: 20.8.7 (in slightly different form)
Permalink:: http://dlmf.nist.gov/28.20.E24
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §28.20(vii), §28.20 and Ch.28

For $s\in\mathbb{Z}$ ,

28.20.25

${\operatorname{M}^{(1)}_{\nu}}\left(z+s\pi\mathrm{i},h\right)=e^{\mathrm{i}s% \pi\nu}{\operatorname{M}^{(1)}_{\nu}}\left(z,h\right),$

${\operatorname{M}^{(2)}_{\nu}}\left(z+s\pi\mathrm{i},h\right)=e^{-\mathrm{i}s% \pi\nu}{\operatorname{M}^{(2)}_{\nu}}\left(z,h\right)+2\mathrm{i}\cot\left(\pi% \nu\right)\sin\left(s\pi\nu\right){\operatorname{M}^{(1)}_{\nu}}\left(z,h% \right),$

${\operatorname{M}^{(3)}_{\nu}}\left(z+s\pi\mathrm{i},h\right)=-\dfrac{\sin% \left({(s-1)\pi\nu}\right)}{\sin\left(\pi\nu\right)}{\operatorname{M}^{(3)}_{% \nu}}\left(z,h\right)-e^{-\mathrm{i}\pi\nu}\frac{\sin\left(s\pi\nu\right)}{% \sin\left(\pi\nu\right)}{\operatorname{M}^{(4)}_{\nu}}\left(z,h\right),$

${\operatorname{M}^{(4)}_{\nu}}\left(z+s\pi\mathrm{i},h\right)=e^{\mathrm{i}\pi% \nu}\dfrac{\sin\left(s\pi\nu\right)}{\sin\left(\pi\nu\right)}{\operatorname{M}% ^{(3)}_{\nu}}\left(z,h\right)+\frac{\sin\left((s+1)\pi\nu\right)}{\sin\left(% \pi\nu\right)}{\operatorname{M}^{(4)}_{\nu}}\left(z,h\right).$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cot\NVar{z}$ : cotangent function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, ${\operatorname{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function, $\sin\NVar{z}$ : sine function, : parameter, : complex variable and $\nu$ : complex parameter
Referenced by:: §28.20(vii)
Permalink:: http://dlmf.nist.gov/28.20.E25
Encodings:: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png
See also:: Annotations for §28.20(vii), §28.20 and Ch.28

When $\nu$ is an integer the right-hand sides of (28.20.25) are replaced by the their limiting values. And for the corresponding identities for the radial functions use (28.20.15) and (28.20.16).

28.19 Expansions in Series of $\operatorname{me}_{\nu+2n}$ Functions 28.21 Graphics

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§28.20 Definitions and Basic Properties

Contents

§28.20(i) Modified Mathieu’s Equation

§28.20(ii) Solutions , , , ,

§28.20(iii) Solutions

§28.20(iv) Radial Mathieu Functions ,

§28.20(v) Solutions , , ,