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

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Keywords:: connection formulas, modified Mathieu functions
Permalink:: http://dlmf.nist.gov/28.22
See also:: Annotations for Ch.28

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

§28.22(i) Integer $\nu$

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Keywords:: joining factors, modified Mathieu functions, radial Mathieu functions
Notes:: See Meixner and Schäfke (1954, §2.76 and p. 214).
Permalink:: http://dlmf.nist.gov/28.22.i
See also:: Annotations for §28.22 and Ch.28

28.22.1 ${\operatorname{Mc}^{(1)}_{m}}\left(z,h\right)=\sqrt{\dfrac{2}{\pi}}\dfrac{1}{g% _{\mathit{e},m}(h)\operatorname{ce}_{m}\left(0,h^{2}\right)}\operatorname{Ce}_% {m}\left(z,h^{2}\right),$

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Symbols:: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{Ce}_{\NVar{\nu}}\left(\NVar{z},\NVar{q}\right)$ : modified Mathieu function, ${\operatorname{Mc}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function, : integer, : parameter, : complex variable and $g_{\mathit{e},n}(h)$ , $g_{\mathit{o},n}(h)$ : joining factors
A&S Ref:: 20.6.15 (in different form)
Permalink:: http://dlmf.nist.gov/28.22.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §28.22(i), §28.22 and Ch.28

28.22.2 ${\operatorname{Ms}^{(1)}_{m}}\left(z,h\right)=\sqrt{\dfrac{2}{\pi}}\frac{1}{g_% {\mathit{o},m}(h)\operatorname{se}_{m}'\left(0,h^{2}\right)}\operatorname{Se}_% {m}\left(z,h^{2}\right),$

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Symbols:: $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, ${\operatorname{Ms}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function, $\operatorname{Se}_{\NVar{\nu}}\left(\NVar{z},\NVar{q}\right)$ : modified Mathieu function, : integer, : parameter, : complex variable and $g_{\mathit{e},n}(h)$ , $g_{\mathit{o},n}(h)$ : joining factors
Permalink:: http://dlmf.nist.gov/28.22.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §28.22(i), §28.22 and Ch.28

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

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Symbols:: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{Ce}_{\NVar{\nu}}\left(\NVar{z},\NVar{q}\right)$ : modified Mathieu function, $\operatorname{Fe}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : modified Mathieu function, ${\operatorname{Mc}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function, : integer, : parameter, : complex variable, $g_{\mathit{e},n}(h)$ , $g_{\mathit{o},n}(h)$ : joining factors, $f_{\mathit{e},n}(h)$ , $f_{\mathit{o},n}(h)$ : joining factors and $C_{n}(q)$ : factor
Permalink:: http://dlmf.nist.gov/28.22.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §28.22(i), §28.22 and Ch.28

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

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Symbols:: $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{Ge}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : modified Mathieu function, ${\operatorname{Ms}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function, $\operatorname{Se}_{\NVar{\nu}}\left(\NVar{z},\NVar{q}\right)$ : modified Mathieu function, : integer, : parameter, : complex variable, $g_{\mathit{e},n}(h)$ , $g_{\mathit{o},n}(h)$ : joining factors, $f_{\mathit{e},n}(h)$ , $f_{\mathit{o},n}(h)$ : joining factors and $S_{n}(q)$ : factor
Permalink:: http://dlmf.nist.gov/28.22.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §28.22(i), §28.22 and Ch.28

The joining factors in the above formulas are given by

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

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Symbols:: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, : 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
See also:: Annotations for §28.22(i), §28.22 and Ch.28

28.22.6 $g_{\mathit{e},2m+1}(h)=(-1)^{m+1}\sqrt{\frac{2}{\pi}}\dfrac{\operatorname{ce}_% {2m+1}'\left(\frac{1}{2}\pi,h^{2}\right)}{hA_{1}^{2m+1}(h^{2})},$

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Symbols:: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, : 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
See also:: Annotations for §28.22(i), §28.22 and Ch.28

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

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Symbols:: $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, : 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
See also:: Annotations for §28.22(i), §28.22 and Ch.28

28.22.8 $g_{\mathit{o},2m+2}(h)=(-1)^{m+1}\sqrt{\dfrac{2}{\pi}}\dfrac{\operatorname{se}% _{2m+2}'\left(\frac{1}{2}\pi,h^{2}\right)}{h^{2}B_{2}^{2m+2}(h^{2})},$

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Symbols:: $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, : 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
See also:: Annotations for §28.22(i), §28.22 and Ch.28

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

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, ${\operatorname{Mc}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : 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
See also:: Annotations for §28.22(i), §28.22 and Ch.28

28.22.10 $f_{\mathit{o},m}(h)=-\sqrt{\ifrac{\pi}{2}}g_{\mathit{o},m}(h){\operatorname{Ms% }^{(2)}_{m}}'\left(0,h\right),$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, ${\operatorname{Ms}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : 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
See also:: Annotations for §28.22(i), §28.22 and Ch.28

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

${\operatorname{Mc}^{(2)}_{m}}'\left(0,h\right)=\sqrt{\ifrac{2}{\pi}}g_{\mathit% {e},m}(h),$

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

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, ${\operatorname{Mc}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function, ${\operatorname{Ms}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function, : 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
See also:: Annotations for §28.22(i), §28.22 and Ch.28

28.22.12

$\operatorname{fe}_{m}'\left(0,h^{2}\right)=\tfrac{1}{2}\pi C_{m}(h^{2})\left(g% _{\mathit{e},m}(h)\right)^{2}\operatorname{ce}_{m}\left(0,h^{2}\right),$

$\operatorname{ge}_{m}\left(0,h^{2}\right)=\tfrac{1}{2}\pi S_{m}(h^{2})\left(g_% {\mathit{o},m}(h)\right)^{2}\operatorname{se}_{m}'\left(0,h^{2}\right).$

§28.22(ii) Noninteger $\nu$

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Keywords:: connection formulas, modified Mathieu functions
Notes:: See Meixner and Schäfke (1954, §2.65).
Permalink:: http://dlmf.nist.gov/28.22.ii
See also:: Annotations for §28.22 and Ch.28

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

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Symbols:: $\operatorname{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, ${\operatorname{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function, $\operatorname{Me}_{\NVar{\nu}}\left(\NVar{z},\NVar{q}\right)$ : modified Mathieu function, : parameter, : complex variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/28.22.E13
Encodings:: TeX, pMML, png
See also:: Annotations for §28.22(ii), §28.22 and Ch.28

Here $\operatorname{me}_{\nu}\left(0,h^{2}\right)$ $(\neq 0)$ is given by (28.14.1) with , and ${\operatorname{M}^{(1)}_{\nu}}\left(0,h\right)$ 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 ${\operatorname{M}^{(2)}_{\nu}}\left(z,h\right)=\cot\left(\nu\pi\right){% \operatorname{M}^{(1)}_{\nu}}\left(z,h\right)-\frac{1}{\sin\left(\nu\pi\right)% }{\operatorname{M}^{(1)}_{-\nu}}\left(z,h\right).$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cot\NVar{z}$ : cotangent function, ${\operatorname{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function, $\sin\NVar{z}$ : sine function, : parameter, : complex variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/28.22.E14
Encodings:: TeX, pMML, png
See also:: Annotations for §28.22(ii), §28.22 and Ch.28

§28.22 Connection Formulas

Contents

§28.22(i) Integer

§28.22(ii) Noninteger

§28.22(i) Integer $\nu$

§28.22(ii) Noninteger $\nu$