reflection properties in z

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11: 11.9 Lommel Functions

… ►where $A$, $B$ are arbitrary constants, $s_{\mu ,\nu }\left(z\right)$ is the Lommel function defined by … ►Another solution of (11.9.1) that is defined for all values of $\mu$ and $\nu$ is $S_{\mu ,\nu }\left(z\right)$, where … ►

Reflection Formulas

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§11.9(ii) Expansions in Series of Bessel Functions

… ►For descriptive properties of $s_{\mu ,\nu }\left(x\right)$ see Steinig (1972). …

12: 28.4 Fourier Series

… ►The Fourier series of the periodic Mathieu functions converge absolutely and uniformly on all compact sets in the $z$-plane. … ►

28.4.1 $${\mathrm{ce}}_{2n}(z,q)=\sum _{m=0}^{\mathrm{\infty }}{A}_{2m}^{2n}(q)\cos 2mz,$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cos z$: cosine function, $m$: integer, $q=h^2$: parameter, $n$: integer, $z$: complex variable and $A_m(q)$: Fourier coefficient

A&S Ref:

20.2.3 (in slightly different form) 20.2.4 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.4(i), §28.4 and Ch.28

… ►Ambiguities in sign are resolved by (28.4.13)–(28.4.16) when $q=0$, and by continuity for the other values of $q$. … ►

§28.4(v) Change of Sign of $q$

… ►

28.4.24 $$\frac{A_{2m}^{2n}(q)}{A_0^{2n}(q)}=\frac{{(-1)}^m}{{(m!)}^2}{\left(\frac{q}{4}\right)}^m\frac{\pi \left(1+O\left(m^{-1}\right)\right)}{w_{\text{II}}(\frac{1}{2}\pi ;a_{2n}\left(q\right),q)},$$

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Symbols:

$O\left(x\right)$: order not exceeding, $a_n\left(q\right)$: eigenvalues of Mathieu equation, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $m$: integer, $q=h^2$: parameter, $n$: integer, $w_{\text{II}}(z;a,q)$: fundamental solution and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.4.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §28.4(vii), §28.4 and Ch.28

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13: 12.2 Differential Equations

… ►In $ℂ$, for $j=0,1,2,3$, $U({(-1)}^{j-1}a,{(-\mathrm{i})}^{j-1}z)$ and $U({(-1)}^{j}a,{(-\mathrm{i})}^{j}z)$ comprise a numerically satisfactory pair of solutions in the half-plane $\frac{1}{4}(2j-3)\pi \le \mathrm{ph}z\le \frac{1}{4}(2j+1)\pi$. ►

§12.2(ii) Values at $z=0$

… ►

§12.2(iv) Reflection Formulas

… ►Properties of $\overline{U}(a,x)$ follow immediately from those of $V(a,x)$ via (12.2.21). … ►For properties of the modulus and phase functions, including differential equations, see Miller (1955, pp. 72–73). …

14: Bibliography F

… ►

Tablicy značeniĭ funkcii $w(z)=e^{-z^2}(1+\frac{2i}{\sqrt{\pi }}{\int }_0^z{e}^{t^2}𝑑t)$ ot kompleksnogo argumenta. Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow (Russian).

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Notes:

In Russian. English translation, see Faddeyeva and Terent’ev (1961)

External Links:

MathReview (John Todd), ZentralBlatt

Referenced by:

§7.21.

Encodings:

BibTeX

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Complex roots of $\sin z=az$, $\cos z=az$, and $\cosh z=az$ . Math. Comp. 30 (135), pp. 541–545.

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External Links:

FullText, MathReview (Gerhard Merz), ZentralBlatt

Referenced by:

§4.46.

Encodings:

BibTeX

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The transformation properties of the sixth Painlevé equation and one-parameter families of solutions. Lett. Nuovo Cimento (2) 30 (17), pp. 539–544.

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External Links:

ISSN 0024-1318, MathReview (D. A. Lutz)

Referenced by:

§32.10(vi).

Encodings:

BibTeX

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Transverse cusp diffraction catastrophes produced by the reflection of ultrasonic tone bursts from a curved surface in water. J. Acoust. Soc. Amer. 92 (5), pp. 2869–2877.

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External Links:

Document

Referenced by:

§36.14(iv).

Encodings:

BibTeX

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Travel time surface of a transverse cusp caustic produced by reflection of acoustical transients from a curved metal surface. J. Acoust. Soc. Amer. 95 (2), pp. 650–660.

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External Links:

Document

Referenced by:

§36.14(iv).

Encodings:

BibTeX

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