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

§28.20 Definitions and Basic Properties

Permalink:: http://dlmf.nist.gov/28.20

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

§28.20(i) Modified Mathieu’s Equation

Keywords:: modified Mathieu functions, modified Mathieu’s equation
Permalink:: http://dlmf.nist.gov/28.20.i

When is replaced by $\pm iz$ , (28.2.1) becomes the modified Mathieu’s equation:

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

Symbols:: $\mathop{\cosh\/}\nolimits z$ : 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), §28.33(i), §28.33(i), §28.34(iv), ¶ ‣ §28.8(iv), ¶ ‣ §28.8(iv)
Permalink:: http://dlmf.nist.gov/28.20.E1
Encodings:: TeX, pMML, png

with its algebraic form

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

Symbols:: $\mathop{\cosh\/}\nolimits z$ : 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

§28.20(ii) Solutions $\mathop{\mathrm{Ce}_{{\nu}}\/}\nolimits$ , $\mathop{\mathrm{Se}_{{\nu}}\/}\nolimits$ , $\mathop{\mathrm{Me}_{{\nu}}\/}\nolimits$ , $\mathop{\mathrm{Fe}_{{n}}\/}\nolimits$ , $\mathop{\mathrm{Ge}_{{n}}\/}\nolimits$

Notes:: See Meixner and Schäfke (1954, §§2.29 and 2.73).
Keywords:: Mathieu functions, modified Mathieu functions
Permalink:: http://dlmf.nist.gov/28.20.ii

28.20.3 $\mathop{\mathrm{Ce}_{{\nu}}\/}\nolimits\!\left(z,q\right)=\mathop{\mathrm{ce}_% {{\nu}}\/}\nolimits\!\left(\pm iz,q\right),$ $\nu\neq-1,-2,\dots$ ,

Defines:: $\mathop{\mathrm{Ce}_{{\nu}}\/}\nolimits\!\left(z,q\right)$ : modified Mathieu function
Symbols:: $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $q=h^{2}$ : 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

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

Defines:: $\mathop{\mathrm{Se}_{{\nu}}\/}\nolimits\!\left(z,q\right)$ : modified Mathieu function
Symbols:: $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $q=h^{2}$ : 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

28.20.5 $\mathop{\mathrm{Me}_{{\nu}}\/}\nolimits\!\left(z,q\right)=\mathop{\mathrm{me}_% {{\nu}}\/}\nolimits\!\left(-iz,q\right),$

Defines:: $\mathop{\mathrm{Me}_{{\nu}}\/}\nolimits\!\left(z,q\right)$ : modified Mathieu function
Symbols:: $\mathop{\mathrm{me}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $q=h^{2}$ : 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

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

Defines:: $\mathop{\mathrm{Fe}_{{n}}\/}\nolimits\!\left(z,q\right)$ : modified Mathieu function
Symbols:: $\mathop{\mathrm{fe}_{{n}}\/}\nolimits\!\left(z,q\right)$ : second solution, Mathieu’s equation, $q=h^{2}$ : 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

28.20.7 $\mathop{\mathrm{Ge}_{{n}}\/}\nolimits\!\left(z,q\right)=\mathop{\mathrm{ge}_{{% n}}\/}\nolimits\!\left(\pm iz,q\right),$ $n=1,2,\dots$ .

Defines:: $\mathop{\mathrm{Ge}_{{n}}\/}\nolimits\!\left(z,q\right)$ : modified Mathieu function
Symbols:: $\mathop{\mathrm{ge}_{{n}}\/}\nolimits\!\left(z,q\right)$ : second solution, Mathieu’s equation, $q=h^{2}$ : 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

§28.20(iii) Solutions $\mathop{{\mathrm{M}^{{(j)}}_{{\nu}}}\/}\nolimits$

Notes:: See Meixner and Schäfke (1954, §2.41).
Defines:: $\mathop{{\mathrm{M}^{{(j)}}_{{\nu}}}\/}\nolimits\!\left(z,h\right)$ : modified Mathieu function
Keywords:: analytic continuation, modified Mathieu functions
Referenced by:: ¶ ‣ §28.8(iv)
Permalink:: http://dlmf.nist.gov/28.20.iii

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

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

Symbols:: $q=h^{2}$ : parameter and : parameter
Permalink:: http://dlmf.nist.gov/28.20.E8
Encodings:: TeX, pMML, png

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

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

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

as $\realpart{z}\to+\infty$ with $-\pi+\delta\leq\imagpart{z}\leq 2\pi-\delta$ , and

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

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

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

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

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, : base of exponential function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\mathrm{sech}\/}\nolimits z$ : hyperbolic secant function, $\imagpart{}$ : imaginary part, $\mathop{{\mathrm{M}^{{(j)}}_{{\nu}}}\/}\nolimits\!\left(z,h\right)$ : 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

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

Symbols:: $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, : base of exponential function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\mathrm{sech}\/}\nolimits z$ : hyperbolic secant function, $\imagpart{}$ : imaginary part, $\mathop{{\mathrm{M}^{{(j)}}_{{\nu}}}\/}\nolimits\!\left(z,h\right)$ : 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

as $\realpart{z}\to+\infty$ with $|\imagpart{z}|\leq\pi-\delta$ .

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

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

Symbols:: $\mathop{{\mathrm{M}^{{(j)}}_{{\nu}}}\/}\nolimits\!\left(z,h\right)$ : 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

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

Symbols:: $\mathop{{\mathrm{M}^{{(j)}}_{{\nu}}}\/}\nolimits\!\left(z,h\right)$ : 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

§28.20(iv) Radial Mathieu Functions $\mathop{{\mathrm{Mc}^{{(j)}}_{{n}}}\/}\nolimits$ , $\mathop{{\mathrm{Ms}^{{(j)}}_{{n}}}\/}\nolimits$

Keywords:: radial Mathieu functions
Permalink:: http://dlmf.nist.gov/28.20.iv

For ,

28.20.15 $\mathop{{\mathrm{Mc}^{{(j)}}_{{n}}}\/}\nolimits\!\left(z,h\right)=\mathop{{% \mathrm{M}^{{(j)}}_{{n}}}\/}\nolimits\!\left(z,h\right),$ $n=0,1,\dots$ ,

Defines:: $\mathop{{\mathrm{Mc}^{{(j)}}_{{n}}}\/}\nolimits\!\left(z,h\right)$ : radial Mathieu function
Symbols:: $\mathop{{\mathrm{M}^{{(j)}}_{{\nu}}}\/}\nolimits\!\left(z,h\right)$ : 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

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

Defines:: $\mathop{{\mathrm{Ms}^{{(j)}}_{{n}}}\/}\nolimits\!\left(z,h\right)$ : radial Mathieu function
Symbols:: $\mathop{{\mathrm{M}^{{(j)}}_{{\nu}}}\/}\nolimits\!\left(z,h\right)$ : 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

§28.20(v) Solutions $\mathop{\mathrm{Ie}_{{n}}\/}\nolimits$ , $\mathop{\mathrm{Io}_{{n}}\/}\nolimits$ , $\mathop{\mathrm{Ke}_{{n}}\/}\nolimits$ , $\mathop{\mathrm{Ko}_{{n}}\/}\nolimits$

Permalink:: http://dlmf.nist.gov/28.20.v

28.20.17 $\mathop{\mathrm{Ie}_{{n}}\/}\nolimits\!\left(z,h\right)=i^{{-n}}\mathop{{% \mathrm{Mc}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z,ih\right),$

Defines:: $\mathop{\mathrm{Ie}_{{n}}\/}\nolimits\!\left(z,h\right)$ : modified Mathieu function
Symbols:: $\mathop{{\mathrm{Mc}^{{(j)}}_{{n}}}\/}\nolimits\!\left(z,h\right)$ : 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

28.20.18 $\mathop{\mathrm{Io}_{{n}}\/}\nolimits\!\left(z,h\right)=i^{{-n}}\mathop{{% \mathrm{Ms}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z,ih\right),$

Defines:: $\mathop{\mathrm{Io}_{{n}}\/}\nolimits\!\left(z,h\right)$ : modified Mathieu function
Symbols:: $\mathop{{\mathrm{Ms}^{{(j)}}_{{n}}}\/}\nolimits\!\left(z,h\right)$ : 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

28.20.19

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

$\mathop{\mathrm{Ke}_{{2m+1}}\/}\nolimits\!\left(z,h\right)=(-1)^{{m+1}}\tfrac{% 1}{2}\pi\mathop{{\mathrm{Mc}^{{(3)}}_{{2m+1}}}\/}\nolimits\!\left(z,ih\right),$

Defines:: $\mathop{\mathrm{Ke}_{{n}}\/}\nolimits\!\left(z,h\right)$ : modified Mathieu function
Symbols:: $\mathop{{\mathrm{Mc}^{{(j)}}_{{n}}}\/}\nolimits\!\left(z,h\right)$ : 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

28.20.20

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

$\mathop{\mathrm{Ko}_{{2m+1}}\/}\nolimits\!\left(z,h\right)=(-1)^{{m+1}}\tfrac{% 1}{2}\pi\mathop{{\mathrm{Ms}^{{(3)}}_{{2m+1}}}\/}\nolimits\!\left(z,ih\right).$

Defines:: $\mathop{\mathrm{Ko}_{{n}}\/}\nolimits\!\left(z,h\right)$ : modified Mathieu function
Symbols:: $\mathop{{\mathrm{Ms}^{{(j)}}_{{n}}}\/}\nolimits\!\left(z,h\right)$ : radial Mathieu function, : integer, : parameter, : integer and : complex variable
Permalink:: http://dlmf.nist.gov/28.20.E20
Encodings:: TeX, TeX, pMML, pMML, png, png

§28.20(vi) Wronskians

Notes:: See Meixner and Schäfke (1954, pp. 170–171).
Keywords:: modified Mathieu functions
Permalink:: http://dlmf.nist.gov/28.20.vi

28.20.21

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

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

Symbols:: $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, $\mathop{{\mathrm{M}^{{(j)}}_{{\nu}}}\/}\nolimits\!\left(z,h\right)$ : modified Mathieu function and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/28.20.E21
Encodings:: TeX, TeX, pMML, pMML, png, png

§28.20(vii) Shift of Variable

Notes:: See Meixner and Schäfke (1954, §2.75 and p. 170).
Keywords:: modified Mathieu functions, radial Mathieu functions
Permalink:: http://dlmf.nist.gov/28.20.vii

28.20.22 $\mathop{{\mathrm{M}^{{(j)}}_{{\nu}}}\/}\nolimits\!\left(z\pm\tfrac{1}{2}\pi i,% h\right)=\mathop{{\mathrm{M}^{{(j)}}_{{\nu}}}\/}\nolimits\!\left(z,\pm ih% \right),$ $\nu\notin\Integer$ .

Symbols:: $\Integer$ : set of all integers, $\mathop{{\mathrm{M}^{{(j)}}_{{\nu}}}\/}\nolimits\!\left(z,h\right)$ : 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

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

28.20.23

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

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

Symbols:: $\mathop{{\mathrm{Mc}^{{(j)}}_{{n}}}\/}\nolimits\!\left(z,h\right)$ : radial Mathieu function, $\mathop{{\mathrm{Ms}^{{(j)}}_{{n}}}\/}\nolimits\!\left(z,h\right)$ : 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

28.20.24

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

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

Symbols:: $\mathop{{\mathrm{Mc}^{{(j)}}_{{n}}}\/}\nolimits\!\left(z,h\right)$ : radial Mathieu function, $\mathop{{\mathrm{Ms}^{{(j)}}_{{n}}}\/}\nolimits\!\left(z,h\right)$ : 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

For $s\in\Integer$ ,

28.20.25

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

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

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

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

Symbols:: $\mathop{\cot\/}\nolimits z$ : cotangent function, : base of exponential function, $\mathop{{\mathrm{M}^{{(j)}}_{{\nu}}}\/}\nolimits\!\left(z,h\right)$ : modified Mathieu function, $\mathop{\sin\/}\nolimits z$ : 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

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).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 28.19 Expansions in Series of $\mathop{\mathrm{me}_{{\nu+2n}}\/}\nolimits$ Functions 28.21 Graphics

§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 , , ,