asymptotic forms for small q

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1: 28.15 Expansions for Small $q$

§28.15 Expansions for Small $q$

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2: 28.4 Fourier Series

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§28.4(vi) Behavior for Small $q$

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3: 28.20 Definitions and Basic Properties

… ►with its algebraic form … ►

28.20.3 $${\mathrm{Ce}}_{\nu }(z,q)={\mathrm{ce}}_{\nu }(\pm \mathrm{i}z,q),$$ $\nu \ne -1,-2,\mathrm{\dots }$,

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Defines:

${\mathrm{Ce}}_{\nu }(z,q)$: modified Mathieu function

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\mathrm{i}$: imaginary unit, $q=h^2$: 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:

pMML, png, TeX

See also:

Annotations for §28.20(ii), §28.20 and Ch.28

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

►Assume first that $\nu$ is real, $q$ is positive, and $a={\lambda }_{\nu }\left(q\right)$; see §28.12(i). …Then from §2.7(ii) it is seen that equation (28.20.2) has independent and unique solutions that are asymptotic to ${\zeta }^{1/2}{\mathrm{e}}^{\pm 2\mathrm{i}h\zeta }$ as $\zeta \to \mathrm{\infty }$ in the respective sectors $|\mathrm{ph}\left(\mp \mathrm{i}\zeta \right)|\le \frac{3}{2}\pi -\delta$, $\delta$ being an arbitrary small positive constant. …