28 Mathieu Functions and Hill’s Equation Modified Mathieu Functions 28.21 Graphics 28.23 Expansions in Series of Bessel Functions

§28.22 Connection Formulas

Keywords:: modified Mathieu functions
Permalink:: http://dlmf.nist.gov/28.22

§28.22(i) Integer $\nu$
§28.22(ii) Noninteger $\nu$

§28.22(i) Integer $\nu$

Notes:: See Meixner and Schäfke (1954, §2.76 and p. 214).
Keywords:: modified Mathieu functions, radial Mathieu functions
Permalink:: http://dlmf.nist.gov/28.22.i

28.22.1 $\mathop{{\mathrm{Mc}^{{(1)}}_{{m}}}\/}\nolimits\!\left(z,h\right)=\sqrt{\dfrac% {2}{\pi}}\dfrac{1}{g_{{\mathit{e},m}}(h)\mathop{\mathrm{ce}_{{m}}\/}\nolimits% \!\left(0,h^{2}\right)}\mathop{\mathrm{Ce}_{{m}}\/}\nolimits\!\left(z,h^{2}% \right),$

Symbols:: $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\mathrm{Ce}_{{\nu}}\/}\nolimits\!\left(z,q\right)$ : modified Mathieu function, $\mathop{{\mathrm{Mc}^{{(j)}}_{{n}}}\/}\nolimits\!\left(z,h\right)$ : radial Mathieu function, : integer, : parameter, $g_{{\mathit{e},n}}(h)$ , $g_{{\mathit{o},n}}(h)$ : joining factors and : complex variable
A&S Ref:: 20.6.15 (in different form)
Permalink:: http://dlmf.nist.gov/28.22.E1
Encodings:: TeX, pMML, png

28.22.2 $\mathop{{\mathrm{Ms}^{{(1)}}_{{m}}}\/}\nolimits\!\left(z,h\right)=\sqrt{\dfrac% {2}{\pi}}\frac{1}{g_{{\mathit{o},m}}(h){\mathop{\mathrm{se}_{{m}}\/}\nolimits^% {{\prime}}}\!\left(0,h^{2}\right)}\mathop{\mathrm{Se}_{{m}}\/}\nolimits\!\left% (z,h^{2}\right),$

Symbols:: $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{{\mathrm{Ms}^{{(j)}}_{{n}}}\/}\nolimits\!\left(z,h\right)$ : radial Mathieu function, $\mathop{\mathrm{Se}_{{\nu}}\/}\nolimits\!\left(z,q\right)$ : modified Mathieu function, : integer, : parameter, $g_{{\mathit{e},n}}(h)$ , $g_{{\mathit{o},n}}(h)$ : joining factors and : complex variable
Permalink:: http://dlmf.nist.gov/28.22.E2
Encodings:: TeX, pMML, png

28.22.3 $\mathop{{\mathrm{Mc}^{{(2)}}_{{m}}}\/}\nolimits\!\left(z,h\right)=\sqrt{\frac{% 2}{\pi}}\dfrac{1}{g_{{\mathit{e},m}}(h)\mathop{\mathrm{ce}_{{m}}\/}\nolimits\!% \left(0,h^{2}\right)}\*\left(-f_{{\mathit{e},m}}(h)\mathop{\mathrm{Ce}_{{m}}\/% }\nolimits\!\left(z,h^{2}\right)+\dfrac{2}{\pi C_{{m}}(h^{2})}\mathop{\mathrm{% Fe}_{{m}}\/}\nolimits\!\left(z,h^{2}\right)\right),$

Symbols:: $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\mathrm{Ce}_{{\nu}}\/}\nolimits\!\left(z,q\right)$ : modified Mathieu function, $\mathop{\mathrm{Fe}_{{n}}\/}\nolimits\!\left(z,q\right)$ : modified Mathieu function, $\mathop{{\mathrm{Mc}^{{(j)}}_{{n}}}\/}\nolimits\!\left(z,h\right)$ : radial Mathieu function, : integer, : parameter, $g_{{\mathit{e},n}}(h)$ , $g_{{\mathit{o},n}}(h)$ : joining factors, $f_{{\mathit{e},n}}(h)$ , $f_{{\mathit{o},n}}(h)$ : joining factors, : complex variable and $C_{n}(q)$ : factor
Permalink:: http://dlmf.nist.gov/28.22.E3
Encodings:: TeX, pMML, png

28.22.4 $\mathop{{\mathrm{Ms}^{{(2)}}_{{m}}}\/}\nolimits\!\left(z,h\right)=\sqrt{\frac{% 2}{\pi}}\dfrac{1}{g_{{\mathit{o},m}}(h){\mathop{\mathrm{se}_{{m}}\/}\nolimits^% {{\prime}}}\!\left(0,h^{2}\right)}\*\left(-f_{{\mathit{o},m}}(h)\mathop{% \mathrm{Se}_{{m}}\/}\nolimits\!\left(z,h^{2}\right)-\dfrac{2}{\pi S_{{m}}(h^{2% })}\mathop{\mathrm{Ge}_{{m}}\/}\nolimits\!\left(z,h^{2}\right)\right).$

Symbols:: $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\mathrm{Ge}_{{n}}\/}\nolimits\!\left(z,q\right)$ : modified Mathieu function, $\mathop{{\mathrm{Ms}^{{(j)}}_{{n}}}\/}\nolimits\!\left(z,h\right)$ : radial Mathieu function, $\mathop{\mathrm{Se}_{{\nu}}\/}\nolimits\!\left(z,q\right)$ : modified Mathieu function, : integer, : parameter, $g_{{\mathit{e},n}}(h)$ , $g_{{\mathit{o},n}}(h)$ : joining factors, $f_{{\mathit{e},n}}(h)$ , $f_{{\mathit{o},n}}(h)$ : joining factors, : complex variable and $S_{n}(q)$ : factor
Permalink:: http://dlmf.nist.gov/28.22.E4
Encodings:: TeX, pMML, png

The joining factors in the above formulas are given by

28.22.5 $g_{{\mathit{e},2m}}(h)=(-1)^{m}\sqrt{\dfrac{2}{\pi}}\dfrac{\mathop{\mathrm{ce}% _{{2m}}\/}\nolimits\!\left(\frac{1}{2}\pi,h^{2}\right)}{A_{0}^{{2m}}(h^{2})},$

Symbols:: $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, : integer, : parameter, $g_{{\mathit{e},n}}(h)$ , $g_{{\mathit{o},n}}(h)$ : joining factors and $A_{{m}}(q)$ : Fourier coefficient
Permalink:: http://dlmf.nist.gov/28.22.E5
Encodings:: TeX, pMML, png

28.22.6 $g_{{\mathit{e},2m+1}}(h)=(-1)^{{m+1}}\sqrt{\frac{2}{\pi}}\dfrac{{\mathop{% \mathrm{ce}_{{2m+1}}\/}\nolimits^{{\prime}}}\!\left(\frac{1}{2}\pi,h^{2}\right% )}{hA_{1}^{{2m+1}}(h^{2})},$

Symbols:: $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, : integer, : parameter, $g_{{\mathit{e},n}}(h)$ , $g_{{\mathit{o},n}}(h)$ : joining factors and $A_{{m}}(q)$ : Fourier coefficient
Permalink:: http://dlmf.nist.gov/28.22.E6
Encodings:: TeX, pMML, png

28.22.7 $g_{{\mathit{o},2m+1}}(h)=(-1)^{m}\sqrt{\dfrac{2}{\pi}}\dfrac{\mathop{\mathrm{% se}_{{2m+1}}\/}\nolimits\!\left(\frac{1}{2}\pi,h^{2}\right)}{hB_{1}^{{2m+1}}(h% ^{2})},$

Symbols:: $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, : integer, : parameter, $g_{{\mathit{e},n}}(h)$ , $g_{{\mathit{o},n}}(h)$ : joining factors and $B_{{m}}(q)$ : Fourier coefficient
Permalink:: http://dlmf.nist.gov/28.22.E7
Encodings:: TeX, pMML, png

28.22.8 $g_{{\mathit{o},2m+2}}(h)=(-1)^{{m+1}}\sqrt{\dfrac{2}{\pi}}\dfrac{{\mathop{% \mathrm{se}_{{2m+2}}\/}\nolimits^{{\prime}}}\!\left(\frac{1}{2}\pi,h^{2}\right% )}{h^{2}B_{2}^{{2m+2}}(h^{2})},$

Symbols:: $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, : integer, : parameter, $g_{{\mathit{e},n}}(h)$ , $g_{{\mathit{o},n}}(h)$ : joining factors and $B_{{m}}(q)$ : Fourier coefficient
Permalink:: http://dlmf.nist.gov/28.22.E8
Encodings:: TeX, pMML, png

28.22.9 $f_{{\mathit{e},m}}(h)=-\sqrt{\ifrac{\pi}{2}}g_{{\mathit{e},m}}(h)\mathop{{% \mathrm{Mc}^{{(2)}}_{{m}}}\/}\nolimits\!\left(0,h\right),$

Symbols:: $\mathop{{\mathrm{Mc}^{{(j)}}_{{n}}}\/}\nolimits\!\left(z,h\right)$ : radial Mathieu function, : integer, : parameter, $g_{{\mathit{e},n}}(h)$ , $g_{{\mathit{o},n}}(h)$ : joining factors and $f_{{\mathit{e},n}}(h)$ , $f_{{\mathit{o},n}}(h)$ : joining factors
A&S Ref:: 20.8.12 (in slightly different form)
Permalink:: http://dlmf.nist.gov/28.22.E9
Encodings:: TeX, pMML, png

28.22.10 $f_{{\mathit{o},m}}(h)=-\sqrt{\ifrac{\pi}{2}}g_{{\mathit{o},m}}(h){\mathop{{% \mathrm{Ms}^{{(2)}}_{{m}}}\/}\nolimits^{{\prime}}}\!\left(0,h\right),$

Symbols:: $\mathop{{\mathrm{Ms}^{{(j)}}_{{n}}}\/}\nolimits\!\left(z,h\right)$ : radial Mathieu function, : integer, : parameter, $g_{{\mathit{e},n}}(h)$ , $g_{{\mathit{o},n}}(h)$ : joining factors and $f_{{\mathit{e},n}}(h)$ , $f_{{\mathit{o},n}}(h)$ : joining factors
A&S Ref:: 20.8.12 (in slightly different form)
Permalink:: http://dlmf.nist.gov/28.22.E10
Encodings:: TeX, pMML, png

where $A_{n}^{m}(h^{2})$ , $B_{n}^{m}(h^{2})$ are as in §28.4(i), and $C_{m}(h^{2})$ , $S_{m}(h^{2})$ are as in §28.5(i). Furthermore,

28.22.11

${\mathop{{\mathrm{Mc}^{{(2)}}_{{m}}}\/}\nolimits^{{\prime}}}\!\left(0,h\right)% =\sqrt{\ifrac{2}{\pi}}g_{{\mathit{e},m}}(h),$

$\mathop{{\mathrm{Ms}^{{(2)}}_{{m}}}\/}\nolimits\!\left(0,h\right)=-\sqrt{% \ifrac{2}{\pi}}g_{{\mathit{o},m}}(h),$

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, : integer, : parameter and $g_{{\mathit{e},n}}(h)$ , $g_{{\mathit{o},n}}(h)$ : joining factors
A&S Ref:: 20.8.12
Permalink:: http://dlmf.nist.gov/28.22.E11
Encodings:: TeX, TeX, pMML, pMML, png, png

28.22.12

${\mathop{\mathrm{fe}_{{m}}\/}\nolimits^{{\prime}}}\!\left(0,h^{2}\right)=% \tfrac{1}{2}\pi C_{m}(h^{2})\left(g_{{\mathit{e},m}}(h)\right)^{2}\mathop{% \mathrm{ce}_{{m}}\/}\nolimits\!\left(0,h^{2}\right),$

$\mathop{\mathrm{ge}_{{m}}\/}\nolimits\!\left(0,h^{2}\right)=\tfrac{1}{2}\pi S_% {m}(h^{2})\left(g_{{\mathit{o},m}}(h)\right)^{2}{\mathop{\mathrm{se}_{{m}}\/}% \nolimits^{{\prime}}}\!\left(0,h^{2}\right).$

Symbols:: $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\mathrm{fe}_{{n}}\/}\nolimits\!\left(z,q\right)$ : second solution, Mathieu’s equation, $\mathop{\mathrm{ge}_{{n}}\/}\nolimits\!\left(z,q\right)$ : second solution, Mathieu’s equation, $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, : integer, : parameter, $g_{{\mathit{e},n}}(h)$ , $g_{{\mathit{o},n}}(h)$ : joining factors, $C_{n}(q)$ : factor and $S_{n}(q)$ : factor
Referenced by:: ¶ ‣ §28.5(i)
Permalink:: http://dlmf.nist.gov/28.22.E12
Encodings:: TeX, TeX, pMML, pMML, png, png

§28.22(ii) Noninteger $\nu$

Notes:: See Meixner and Schäfke (1954, §2.65).
Keywords:: modified Mathieu functions
Permalink:: http://dlmf.nist.gov/28.22.ii

28.22.13 $\mathop{{\mathrm{M}^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z,h\right)=\frac{% \mathop{{\mathrm{M}^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(0,h\right)}{\mathop{% \mathrm{me}_{{\nu}}\/}\nolimits\!\left(0,h^{2}\right)}\mathop{\mathrm{Me}_{{% \nu}}\/}\nolimits\!\left(z,h^{2}\right).$

Symbols:: $\mathop{\mathrm{me}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{{\mathrm{M}^{{(j)}}_{{\nu}}}\/}\nolimits\!\left(z,h\right)$ : modified Mathieu function, $\mathop{\mathrm{Me}_{{\nu}}\/}\nolimits\!\left(z,q\right)$ : modified Mathieu function, : parameter, : complex variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/28.22.E13
Encodings:: TeX, pMML, png

Here $\mathop{\mathrm{me}_{{\nu}}\/}\nolimits\!\left(0,h^{2}\right)$ $(\neq 0)$ is given by (28.14.1) with , and $\mathop{{\mathrm{M}^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(0,h\right)$ is given by (28.24.1) with , , and chosen so that $|c_{{2n}}^{{\nu}}(h^{2})|=\max(|c_{{2\ell}}^{{\nu}}(h^{2})|)$ , where the maximum is taken over all integers $\ell$ .

28.22.14 $\mathop{{\mathrm{M}^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z,h\right)=\mathop{% \cot\/}\nolimits\!\left(\nu\pi\right)\mathop{{\mathrm{M}^{{(1)}}_{{\nu}}}\/}% \nolimits\!\left(z,h\right)-\frac{1}{\mathop{\sin\/}\nolimits\!\left(\nu\pi% \right)}\mathop{{\mathrm{M}^{{(1)}}_{{-\nu}}}\/}\nolimits\!\left(z,h\right).$

Symbols:: $\mathop{\cot\/}\nolimits z$ : cotangent 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
Permalink:: http://dlmf.nist.gov/28.22.E14
Encodings:: TeX, pMML, png

§28.22 Connection Formulas

Contents

§28.22(i) Integer

§28.22(ii) Noninteger

§28.22(i) Integer $\nu$

§28.22(ii) Noninteger $\nu$