reflection properties in ν

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1: 28.12 Definitions and Basic Properties

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28.12.2 $${\lambda }_{\nu }\left(-q\right)={\lambda }_{\nu }\left(q\right)={\lambda }_{-\nu }\left(q\right).$$

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Symbols:

${\lambda }_{\nu +2n}\left(q\right)$: eigenvalues of Mathieu equation, $q=h^2$: parameter and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.12.E2

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Annotations for §28.12(i), §28.12 and Ch.28

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28.12.10 $$\overline{{\mathrm{me}}_{\nu }(z,q)}={\mathrm{me}}_{\overline{\nu }}(-\overline{z},\overline{q}).$$

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\overline{z}$: complex conjugate, $q=h^2$: parameter, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§28.12(ii)

Permalink:

http://dlmf.nist.gov/28.12.E10

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Annotations for §28.12(ii), §28.12 and Ch.28

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28.12.15 $${\mathrm{se}}_{\nu }(z,q)=-{\mathrm{se}}_{\nu }(-z,q)=-{\mathrm{se}}_{-\nu }(z,q).$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $q=h^2$: parameter, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.12.E15

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See also:

Annotations for §28.12(iii), §28.12 and Ch.28

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2: 28.5 Second Solutions ${\mathrm{fe}}_n$, ${\mathrm{ge}}_n$

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$$S_{2m+2}(-q)=S_{2m+2}(q).$$

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3: 28.2 Definitions and Basic Properties

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Change of Sign of $q$

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28.2.37 $${\mathrm{se}}_{2n+2}(z,-q)={(-1)}^n{\mathrm{se}}_{2n+2}(\frac{1}{2}\pi -z,q).$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $q=h^2$: parameter, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/28.2.E37

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Annotations for §28.2(vi), §28.2 and Ch.28

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4: 28.4 Fourier Series

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§28.4(v) Change of Sign of $q$

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5: 31.8 Solutions via Quadratures

… ►The curve $\mathrm{\Gamma }$ reflects the finite-gap property of Equation (31.2.1) when the exponent parameters satisfy (31.8.1) for $m_j\in \mathbb{Z}$. …

6: 10.68 Modulus and Phase Functions

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§10.68(ii) Basic Properties

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${\varphi }_{-\nu }\left(x\right)={\varphi }_{\nu }\left(x\right)+\nu \pi .$

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§10.68(iv) Further Properties

►Additional properties of the modulus and phase functions are given in Young and Kirk (1964, pp. xi–xv). …

7: 28.31 Equations of Whittaker–Hill and Ince

… ►It has been discussed in detail by Arscott (1967) for , and by Urwin and Arscott (1970) for $k^2>0$. … ►in (28.31.1). … ►and $m=0,1,\mathrm{\dots },n$ in all cases. … ►If $p\to \mathrm{\infty }$ and $\xi \to 0$ in such a way that $p\xi \to 2q$, then in the notation of §§28.2(v) and 28.2(vi) … ►More important are the double orthogonality relations for $p_1\ne p_2$ or $m_1\ne m_2$ or both, given by …

8: 10.61 Definitions and Basic Properties

§10.61 Definitions and Basic Properties

… ►Most properties of ${\mathrm{ber}}_{\nu }x$, ${\mathrm{bei}}_{\nu }x$, ${\ker }_{\nu }x$, and ${\mathrm{kei}}_{\nu }x$ follow straightforwardly from the above definitions and results given in preceding sections of this chapter. … ►

§10.61(iii) Reflection Formulas for Arguments

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§10.61(iv) Reflection Formulas for Orders

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9: 11.9 Lommel Functions

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Reflection Formulas

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§11.9(ii) Expansions in Series of Bessel Functions

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11.9.7 $$s_{\mu ,\nu }\left(z\right)=2^{\mu +1}\sum _{k=0}^{\mathrm{\infty }}\frac{(2k+\mu +1)\mathrm{\Gamma }\left(k+\mu +1\right)}{k!(2k+\mu -\nu +1)(2k+\mu +\nu +1)}{J}_{2k+\mu +1}\left(z\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $s_{\mu ,\nu }\left(z\right)$: Lommel function, $!$: factorial (as in $n!$), $z$: complex variable, $\nu$: real or complex order and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/11.9.E7

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See also:

Annotations for §11.9(ii), §11.9 and Ch.11

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11.9.8 $$s_{\mu ,\nu }\left(z\right)=2^{(\mu +\nu -1)/2}\mathrm{\Gamma }\left(\frac{1}{2}\mu +\frac{1}{2}\nu +\frac{1}{2}\right)z^{(\mu +1-\nu )/2}\sum _{k=0}^{\mathrm{\infty }}\frac{{(\frac{1}{2}z)}^k}{k!(2k+\mu -\nu +1)}{J}_{k+\frac{1}{2}(\mu +\nu +1)}\left(z\right).$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $s_{\mu ,\nu }\left(z\right)$: Lommel function, $!$: factorial (as in $n!$), $z$: complex variable, $\nu$: real or complex order and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/11.9.E8

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Annotations for §11.9(ii), §11.9 and Ch.11

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

10: Bibliography F

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Expansions of hypergeometric functions in hypergeometric functions. Math. Comp. 15 (76), pp. 390–395.

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External Links:

FullText, MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§16.10.

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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Finite Differences and Difference Equations in the Real Domain. Clarendon Press, Oxford.

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External Links:

MathReview (W. J. Trjitzinsky), ZentralBlatt

Referenced by:

§26.1, §26.1, Notations S, Notations S.

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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