28 Mathieu Functions and Hill’s Equation Modified Mathieu Functions 28.19 Expansions in Series of

\operatorname{me}_{\nu+2n}

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

28.20.1		$w^{\prime\prime}-\left(a-2q\cosh\left(2z\right)\right)w=0,$
ⓘ Symbols: $\cosh\NVar{z}$ : hyperbolic cosine function, $q=h^{2}$ : parameter, $z$ : complex variable, $a$ : parameter and $w(z)$ : 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		${(\zeta^{2}-1)w^{\prime\prime}+\zeta w^{\prime}+\left(4q\zeta^{2}-2q-a\right)w% =0},$
$\zeta=\cosh z$ .
ⓘ Symbols: $\cosh\NVar{z}$ : hyperbolic cosine function, $q=h^{2}$ : parameter, $z$ : complex variable, $a$ : parameter and $w(z)$ : 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	$\displaystyle\operatorname{Ce}_{\nu}\left(z,q\right)$	$\displaystyle=\operatorname{ce}_{\nu}\left(\pm\mathrm{i}z,q\right),$
$\nu\neq-1,-2,\dots$ ,
ⓘ 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, $z$ : 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	$\displaystyle\operatorname{Se}_{\nu}\left(z,q\right)$	$\displaystyle=\mp\mathrm{i}\operatorname{se}_{\nu}\left(\pm\mathrm{i}z,q\right),$
$\nu\neq 0,-1,\dots$ ,
ⓘ 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, $z$ : 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	$\displaystyle\operatorname{Me}_{\nu}\left(z,q\right)$	$\displaystyle=\operatorname{me}_{\nu}\left(-\mathrm{i}z,q\right),$
ⓘ 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, $z$ : 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	$\displaystyle\operatorname{Fe}_{n}\left(z,q\right)$	$\displaystyle=\mp\mathrm{i}\operatorname{fe}_{n}\left(\pm\mathrm{i}z,q\right),$
$n=0,1,\dots$ ,
ⓘ 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, $n$ : integer and $z$ : 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	$\displaystyle\operatorname{Ge}_{n}\left(z,q\right)$	$\displaystyle=\operatorname{ge}_{n}\left(\pm\mathrm{i}z,q\right),$
$n=1,2,\dots$ .
ⓘ 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, $n$ : integer and $z$ : 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, $q$ is positive, and $a=\lambda_{\nu}\left(q\right)$ ; see §28.12(i). Write

28.20.8		$h=\sqrt{q}\;(>0).$
ⓘ Symbols: $q=h^{2}$ : parameter and $h$ : 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		${\operatorname{M}^{(3)}_{\nu}}\left(z,h\right)={H^{(1)}_{\nu}}\left(2h\cosh z% \right)\left(1+O\left(\operatorname{sech}z\right)\right),$
ⓘ 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, $h$ : parameter, $z$ : 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		${\operatorname{M}^{(4)}_{\nu}}\left(z,h\right)={H^{(2)}_{\nu}}\left(2h\cosh z% \right)\left(1+O\left(\operatorname{sech}z\right)\right),$
ⓘ 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, $h$ : parameter, $z$ : 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 $z$ is real and have the properties

28.20.11		${\operatorname{M}^{(1)}_{\nu}}\left(z,h\right)=J_{\nu}\left(2h\cosh z\right)+e% ^{\|\Im\left(2h\cosh z\right)\|}O\left(\left(\operatorname{sech}z\right)^{3/2}% \right),$
ⓘ 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, $h$ : parameter, $z$ : 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		${\operatorname{M}^{(2)}_{\nu}}\left(z,h\right)=Y_{\nu}\left(2h\cosh z\right)+e% ^{\|\Im\left(2h\cosh z\right)\|}O\left((\operatorname{sech}z)^{3/2}\right),$
ⓘ 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, $h$ : parameter, $z$ : 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 $z$ , $h$ , and $\nu$ the functions ${\operatorname{M}^{(j)}_{\nu}}\left(z,h\right)$ , $j=1,2,3,4$ , are determined by analytic continuation. Furthermore,

28.20.13	$\displaystyle{\operatorname{M}^{(3)}_{\nu}}\left(z,h\right)$	$\displaystyle={\operatorname{M}^{(1)}_{\nu}}\left(z,h\right)+\mathrm{i}{% \operatorname{M}^{(2)}_{\nu}}\left(z,h\right),$
ⓘ Symbols: $\mathrm{i}$ : imaginary unit, ${\operatorname{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function, $h$ : parameter, $z$ : 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	$\displaystyle{\operatorname{M}^{(4)}_{\nu}}\left(z,h\right)$	$\displaystyle={\operatorname{M}^{(1)}_{\nu}}\left(z,h\right)-\mathrm{i}{% \operatorname{M}^{(2)}_{\nu}}\left(z,h\right).$
ⓘ Symbols: $\mathrm{i}$ : imaginary unit, ${\operatorname{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function, $h$ : parameter, $z$ : 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 $j=1,2,3,4$ ,

28.20.15	$\displaystyle{\operatorname{Mc}^{(j)}_{n}}\left(z,h\right)$	$\displaystyle={\operatorname{M}^{(j)}_{n}}\left(z,h\right),$
$n=0,1,\dots$ ,
ⓘ 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, $h$ : parameter, $n$ : integer, $j$ : integer and $z$ : 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	$\displaystyle{\operatorname{Ms}^{(j)}_{n}}\left(z,h\right)$	$\displaystyle=(-1)^{n}{\operatorname{M}^{(j)}_{-n}}\left(z,h\right),$
$n=1,2,\dots$ .
ⓘ 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, $h$ : parameter, $n$ : integer, $j$ : integer and $z$ : 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	$\displaystyle\operatorname{Ie}_{n}\left(z,h\right)$	$\displaystyle={\mathrm{i}}^{-n}{\operatorname{Mc}^{(1)}_{n}}\left(z,\mathrm{i}% h\right),$
ⓘ 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, $h$ : parameter, $n$ : integer and $z$ : 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	$\displaystyle\operatorname{Io}_{n}\left(z,h\right)$	$\displaystyle={\mathrm{i}}^{-n}{\operatorname{Ms}^{(1)}_{n}}\left(z,\mathrm{i}% h\right),$
ⓘ 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, $h$ : parameter, $n$ : integer and $z$ : 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	$\displaystyle\operatorname{Ke}_{2m}\left(z,h\right)$	$\displaystyle=(-1)^{m}\tfrac{1}{2}\pi\mathrm{i}{\operatorname{Mc}^{(3)}_{2m}}% \left(z,\mathrm{i}h\right),$
28.20.19	$\displaystyle\operatorname{Ke}_{2m+1}\left(z,h\right)$	$\displaystyle=(-1)^{m+1}\tfrac{1}{2}\pi{\operatorname{Mc}^{(3)}_{2m+1}}\left(z% ,\mathrm{i}h\right),$
ⓘ 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, $m$ : integer, $h$ : parameter, $n$ : integer and $z$ : 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	$\displaystyle\operatorname{Ko}_{2m}\left(z,h\right)$	$\displaystyle=(-1)^{m}\tfrac{1}{2}\pi\mathrm{i}{\operatorname{Ms}^{(3)}_{2m}}% \left(z,\mathrm{i}h\right),$
28.20.20	$\displaystyle\operatorname{Ko}_{2m+1}\left(z,h\right)$	$\displaystyle=(-1)^{m+1}\tfrac{1}{2}\pi{\operatorname{Ms}^{(3)}_{2m+1}}\left(z% ,\mathrm{i}h\right).$
ⓘ 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, $m$ : integer, $h$ : parameter, $n$ : integer and $z$ : 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	$\displaystyle\mathscr{W}\left\{{\operatorname{M}^{(1)}_{\nu}},{\operatorname{M% }^{(2)}_{\nu}}\right\}$	$\displaystyle=-\mathscr{W}\left\{{\operatorname{M}^{(2)}_{\nu}},{\operatorname% {M}^{(3)}_{\nu}}\right\}=-\mathscr{W}\left\{{\operatorname{M}^{(2)}_{\nu}},{% \operatorname{M}^{(4)}_{\nu}}\right\}=\ifrac{2}{\pi},$
28.20.21	$\displaystyle\mathscr{W}\left\{{\operatorname{M}^{(1)}_{\nu}},{\operatorname{M% }^{(3)}_{\nu}}\right\}$	$\displaystyle=-\mathscr{W}\left\{{\operatorname{M}^{(1)}_{\nu}},{\operatorname% {M}^{(4)}_{\nu}}\right\}=-\tfrac{1}{2}\mathscr{W}\left\{{\operatorname{M}^{(3)% }_{\nu}},{\operatorname{M}^{(4)}_{\nu}}\right\}=\ifrac{2\mathrm{i}}{\pi}.$
ⓘ 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		${\operatorname{M}^{(j)}_{\nu}}\left(z\pm\tfrac{1}{2}\pi\mathrm{i},h\right)={% \operatorname{M}^{(j)}_{\nu}}\left(z,\pm\mathrm{i}h\right),$
$\nu\notin\mathbb{Z}$ .
ⓘ 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, $h$ : parameter, $j$ : integer, $z$ : 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	$\displaystyle{\operatorname{Mc}^{(j)}_{2n}}\left(z\pm\tfrac{1}{2}\pi\mathrm{i}% ,h\right)$	$\displaystyle={\operatorname{Mc}^{(j)}_{2n}}\left(z,\pm\mathrm{i}h\right),$
28.20.23	$\displaystyle{\operatorname{Ms}^{(j)}_{2n+1}}\left(z\pm\tfrac{1}{2}\pi\mathrm{% i},h\right)$	$\displaystyle={\operatorname{Mc}^{(j)}_{2n+1}}\left(z,\pm\mathrm{i}h\right),$
ⓘ 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, $h$ : parameter, $n$ : integer, $j$ : integer and $z$ : 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	$\displaystyle{\operatorname{Mc}^{(j)}_{2n+1}}\left(z\pm\tfrac{1}{2}\pi\mathrm{% i},h\right)$	$\displaystyle={\operatorname{Ms}^{(j)}_{2n+1}}\left(z,\pm\mathrm{i}h\right),$
28.20.24	$\displaystyle{\operatorname{Ms}^{(j)}_{2n+2}}\left(z\pm\tfrac{1}{2}\pi\mathrm{% i},h\right)$	$\displaystyle={\operatorname{Ms}^{(j)}_{2n+2}}\left(z,\pm\mathrm{i}h\right).$
ⓘ 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, $h$ : parameter, $n$ : integer, $j$ : integer and $z$ : 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	$\displaystyle{\operatorname{M}^{(1)}_{\nu}}\left(z+s\pi\mathrm{i},h\right)$	$\displaystyle=e^{\mathrm{i}s\pi\nu}{\operatorname{M}^{(1)}_{\nu}}\left(z,h% \right),$
	$\displaystyle{\operatorname{M}^{(2)}_{\nu}}\left(z+s\pi\mathrm{i},h\right)$	$\displaystyle=e^{-\mathrm{i}s\pi\nu}{\operatorname{M}^{(2)}_{\nu}}\left(z,h% \right)+2\mathrm{i}\cot\left(\pi\nu\right)\sin\left(s\pi\nu\right){% \operatorname{M}^{(1)}_{\nu}}\left(z,h\right),$
	$\displaystyle{\operatorname{M}^{(3)}_{\nu}}\left(z+s\pi\mathrm{i},h\right)$	$\displaystyle=-\dfrac{\sin\left({(s-1)\pi\nu}\right)}{\sin\left(\pi\nu\right)}% {\operatorname{M}^{(3)}_{\nu}}\left(z,h\right)-e^{-\mathrm{i}\pi\nu}\frac{\sin% \left(s\pi\nu\right)}{\sin\left(\pi\nu\right)}{\operatorname{M}^{(4)}_{\nu}}% \left(z,h\right),$
	$\displaystyle{\operatorname{M}^{(4)}_{\nu}}\left(z+s\pi\mathrm{i},h\right)$	$\displaystyle=e^{\mathrm{i}\pi\nu}\dfrac{\sin\left(s\pi\nu\right)}{\sin\left(% \pi\nu\right)}{\operatorname{M}^{(3)}_{\nu}}\left(z,h\right)+\frac{\sin\left((% s+1)\pi\nu\right)}{\sin\left(\pi\nu\right)}{\operatorname{M}^{(4)}_{\nu}}\left% (z,h\right).$
ⓘ 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, $h$ : parameter, $z$ : 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