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11: 28.15 Expansions for Small $q$

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28.15.3 $${\mathrm{me}}_{\nu }(z,q)={\mathrm{e}}^{\mathrm{i}\nu z}-\frac{q}{4}\left(\frac{1}{\nu +1}{\mathrm{e}}^{\mathrm{i}(\nu +2)z}-\frac{1}{\nu -1}{\mathrm{e}}^{\mathrm{i}(\nu -2)z}\right)+\frac{q^2}{32}\left(\frac{1}{(\nu +1)(\nu +2)}{\mathrm{e}}^{\mathrm{i}(\nu +4)z}+\frac{1}{(\nu -1)(\nu -2)}{\mathrm{e}}^{\mathrm{i}(\nu -4)z}-\frac{2({\nu }^2+1)}{{({\nu }^2-1)}^2}{\mathrm{e}}^{\mathrm{i}\nu z}\right)+\mathrm{\cdots };$$

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $q=h^2$: parameter, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

20.3.17 (only two terms and without normalization)

Permalink:

http://dlmf.nist.gov/28.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.15(ii), §28.15 and Ch.28

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12: 19.10 Relations to Other Functions

… ►If $y>0$ is assumed (without loss of generality), then …

13: Charles W. Clark

… ►He has been a Visiting Fellow at the Australian National University, a Dr. Lee Fellow at Christ Church College of the University of Oxford, and Visiting Professor at the National University of Singapore. …

14: Leonard C. Maximon

… ►He had been an Assistant Professor in the Graduate Division of Applied Mathematics at Brown University, a Visiting Professor at the Norwegian Technical University in Trondheim, Norway, and a Physicist at the Center for Radiation Research at the National Bureau of Standards. …

15: Wadim Zudilin

… ►He has been an Associate Professor at Moscow State University (Russia), an Ostrowski Fellow at Institut Henri Poincaré and Université Paris 6 (France), a Humboldt Fellow at the Cologne University (Germany) and a Visiting Researcher at the Max Planck Institute for Mathematics in Bonn (Germany). …

16: Guide to Searching the DLMF

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Table 1: Query Examples

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Query	Matching records contain
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int adj sin	$\int$ immediately followed by $\sin$ without any intervening terms.
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17: 14.16 Zeros

… ►For all cases concerning ${𝖯}_{\nu }^{\mu }\left(x\right)$ and $P_{\nu }^{\mu }\left(x\right)$ we assume that $\nu \ge -\frac{1}{2}$ without loss of generality (see (14.9.5) and (14.9.11)). …

18: 21.3 Symmetry and Quasi-Periodicity

… ►Because of this property, the elements of $𝜶$ and $𝜷$ are usually restricted to $[0,1)$, without loss of generality. …

19: 26.3 Lattice Paths: Binomial Coefficients

… ► $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$ is the number of ways of choosing $n$ objects from a collection of $m$ distinct objects without regard to order. …

20: 32.2 Differential Equations

… ►If $\gamma \delta \ne 0$ in ${\text{P}}_{\text{III}}$, then set $\gamma =1$ and $\delta =-1$, without loss of generality, by rescaling $w$ and $z$ if necessary. If $\gamma =0$ and $\alpha \delta \ne 0$ in ${\text{P}}_{\text{III}}$, then set $\alpha =1$ and $\delta =-1$, without loss of generality. Lastly, if $\delta =0$ and $\beta \gamma \ne 0$, then set $\beta =-1$ and $\gamma =1$, without loss of generality. ►If $\delta \ne 0$ in ${\text{P}}_{\text{V}}$, then set $\delta =-\frac{1}{2}$, without loss of generality. …