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 $\nu$

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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	$\displaystyle{\operatorname{Mc}^{(1)}_{m}}\left(z,h\right)$	$\displaystyle=\sqrt{\dfrac{2}{\pi}}\dfrac{1}{g_{\mathit{e},m}(h)\operatorname{% ce}_{m}\left(0,h^{2}\right)}\operatorname{Ce}_{m}\left(z,h^{2}\right),$
ⓘ 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, $m$ : integer, $h$ : parameter, $z$ : 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	$\displaystyle{\operatorname{Ms}^{(1)}_{m}}\left(z,h\right)$	$\displaystyle=\sqrt{\dfrac{2}{\pi}}\frac{1}{g_{\mathit{o},m}(h)\operatorname{% se}_{m}'\left(0,h^{2}\right)}\operatorname{Se}_{m}\left(z,h^{2}\right),$
ⓘ 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, $m$ : integer, $h$ : parameter, $z$ : 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	$\displaystyle{\operatorname{Mc}^{(2)}_{m}}\left(z,h\right)$	$\displaystyle=\sqrt{\frac{2}{\pi}}\dfrac{1}{g_{\mathit{e},m}(h)\operatorname{% ce}_{m}\left(0,h^{2}\right)}\*\left(-f_{\mathit{e},m}(h)\operatorname{Ce}_{m}% \left(z,h^{2}\right)+\dfrac{2}{\pi C_{m}(h^{2})}\operatorname{Fe}_{m}\left(z,h% ^{2}\right)\right),$
ⓘ 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, $m$ : integer, $h$ : parameter, $z$ : 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	$\displaystyle{\operatorname{Ms}^{(2)}_{m}}\left(z,h\right)$	$\displaystyle=\sqrt{\frac{2}{\pi}}\dfrac{1}{g_{\mathit{o},m}(h)\operatorname{% se}_{m}'\left(0,h^{2}\right)}\*\left(-f_{\mathit{o},m}(h)\operatorname{Se}_{m}% \left(z,h^{2}\right)-\dfrac{2}{\pi S_{m}(h^{2})}\operatorname{Ge}_{m}\left(z,h% ^{2}\right)\right).$
ⓘ 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, $m$ : integer, $h$ : parameter, $z$ : 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	$\displaystyle g_{\mathit{e},2m}(h)$	$\displaystyle=(-1)^{m}\sqrt{\dfrac{2}{\pi}}\dfrac{\operatorname{ce}_{2m}\left(% \frac{1}{2}\pi,h^{2}\right)}{A_{0}^{2m}(h^{2})},$
ⓘ 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, $m$ : integer, $h$ : 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	$\displaystyle g_{\mathit{e},2m+1}(h)$	$\displaystyle=(-1)^{m+1}\sqrt{\frac{2}{\pi}}\dfrac{\operatorname{ce}_{2m+1}'% \left(\frac{1}{2}\pi,h^{2}\right)}{hA_{1}^{2m+1}(h^{2})},$
ⓘ 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, $m$ : integer, $h$ : 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	$\displaystyle g_{\mathit{o},2m+1}(h)$	$\displaystyle=(-1)^{m}\sqrt{\dfrac{2}{\pi}}\dfrac{\operatorname{se}_{2m+1}% \left(\frac{1}{2}\pi,h^{2}\right)}{hB_{1}^{2m+1}(h^{2})},$
ⓘ 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, $m$ : integer, $h$ : 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	$\displaystyle g_{\mathit{o},2m+2}(h)$	$\displaystyle=(-1)^{m+1}\sqrt{\dfrac{2}{\pi}}\dfrac{\operatorname{se}_{2m+2}'% \left(\frac{1}{2}\pi,h^{2}\right)}{h^{2}B_{2}^{2m+2}(h^{2})},$
ⓘ 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, $m$ : integer, $h$ : 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	$\displaystyle f_{\mathit{e},m}(h)$	$\displaystyle=-\sqrt{\ifrac{\pi}{2}}g_{\mathit{e},m}(h){\operatorname{Mc}^{(2)% }_{m}}\left(0,h\right),$
ⓘ 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, $m$ : integer, $h$ : 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	$\displaystyle f_{\mathit{o},m}(h)$	$\displaystyle=-\sqrt{\ifrac{\pi}{2}}g_{\mathit{o},m}(h){\operatorname{Ms}^{(2)% }_{m}}'\left(0,h\right),$
ⓘ 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, $m$ : integer, $h$ : 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	$\displaystyle{\operatorname{Mc}^{(2)}_{m}}'\left(0,h\right)$	$\displaystyle=\sqrt{\ifrac{2}{\pi}}g_{\mathit{e},m}(h),$
28.22.11	$\displaystyle{\operatorname{Ms}^{(2)}_{m}}\left(0,h\right)$	$\displaystyle=-\sqrt{\ifrac{2}{\pi}}g_{\mathit{o},m}(h),$
ⓘ 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, $m$ : integer, $h$ : 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	$\displaystyle\operatorname{fe}_{m}'\left(0,h^{2}\right)$	$\displaystyle=\tfrac{1}{2}\pi C_{m}(h^{2})\left(g_{\mathit{e},m}(h)\right)^{2}% \operatorname{ce}_{m}\left(0,h^{2}\right),$
28.22.12	$\displaystyle\operatorname{ge}_{m}\left(0,h^{2}\right)$	$\displaystyle=\tfrac{1}{2}\pi S_{m}(h^{2})\left(g_{\mathit{o},m}(h)\right)^{2}% \operatorname{se}_{m}'\left(0,h^{2}\right).$
ⓘ Symbols: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\operatorname{fe}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : second solution, Mathieu’s equation, $\operatorname{ge}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : second solution, Mathieu’s equation, $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $m$ : integer, $h$ : 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 See also: Annotations for §28.22(i), §28.22 and Ch.28

§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		${\operatorname{M}^{(1)}_{\nu}}\left(z,h\right)=\frac{{\operatorname{M}^{(1)}_{% \nu}}\left(0,h\right)}{\operatorname{me}_{\nu}\left(0,h^{2}\right)}% \operatorname{Me}_{\nu}\left(z,h^{2}\right).$
ⓘ 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, $h$ : parameter, $z$ : 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 $z=0$ , and ${\operatorname{M}^{(1)}_{\nu}}\left(0,h\right)$ is given by (28.24.1) with $j=1$ , $z=0$ , and $n$ 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		${\operatorname{M}^{(2)}_{\nu}}\left(z,h\right)=\cot\left(\nu\pi\right){% \operatorname{M}^{(1)}_{\nu}}\left(z,h\right)-\frac{1}{\sin\left(\nu\pi\right)% }{\operatorname{M}^{(1)}_{-\nu}}\left(z,h\right).$
ⓘ 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, $h$ : parameter, $z$ : 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 ν

§28.22(ii) Noninteger ν

§28.22(i) Integer $\nu$

§28.22(ii) Noninteger $\nu$