§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 $z$ 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, $z$ : 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), §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, $z$ : 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

§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, $z$ : 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

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, $z$ : 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

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, $z$ : 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

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, $n$ : 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

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, $n$ : 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

§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, $q$ 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 $h$ : 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, $h$ : parameter, $z$ : 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, $h$ : parameter, $z$ : 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 $z$ 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, $e$ : 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, $h$ : 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

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, $e$ : 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, $h$ : 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

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

For other values of $z$ , $h,$ and $\nu$ the functions $\mathop{{\mathrm{M}^{{(j)}}_{{\nu}}}\/}\nolimits\!\left(z,h\right)$ , $j=1,2,3,4,$ 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, $h$ : 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

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, $h$ : 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

§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 $j=1,2,3,4$ ,

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, $h$ : 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

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, $h$ : 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

§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, $h$ : 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

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, $h$ : 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

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, $m$ : 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

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, $m$ : integer, $h$ : parameter, $n$ : integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/28.20.E20
Encodings:: TeX, TeX, pMML, pMML, png, png

§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, $h$ : parameter, $j$ : integer, $z$ : 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, $h$ : 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

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, $h$ : 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

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

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).

§28.20 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 , , ,