reflection properties

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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

Encodings:

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See also:

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

Encodings:

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See also:

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

Encodings:

pMML, png, TeX

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

Encodings:

pMML, png, TeX

See also:

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: 28.31 Equations of Whittaker–Hill and Ince

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$${\mathrm{ℎ𝑠}}_{2n+2}^{2m+2}(z,-\xi )={(-1)}^m{\mathrm{ℎ𝑠}}_{2n+2}^{2m+2}(\frac{1}{2}\pi -z,\xi ).$$

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6: 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 $m_j\in \mathbb{Z}$. …

7: 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). …

8: 37.19 Other Orthogonal Polynomials of $d$ Variables

… ►Let $R_{+}$ be the set of positive roots and let $𝐯\mapsto {\kappa }_{𝐯}$ be a nonnegative function defined on $R_{+}$ with the property that it takes constant value in each conjugacy class of roots. …

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10: 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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