reflection properties in z

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1: 28.12 Definitions and Basic Properties

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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:

pMML, png, TeX

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: 28.2 Definitions and Basic Properties

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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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3: 28.5 Second Solutions ${\mathrm{fe}}_n$, ${\mathrm{ge}}_n$

… ►The functions $f_n(z,q)$, $g_n(z,q)$ are unique. … ►As $q\to 0$ with $n\ne 0$, $C_n(q)\to 0$, $S_n(q)\to 0$, $C_n(q)f_n(z,q)\to \sin nz$, and $S_n(q)g_n(z,q)\to \cos nz$. …(Other normalizations for $C_n(q)$ and $S_n(q)$ can be found in the literature, but most formulas—including connection formulas—are unaffected since ${\mathrm{fe}}_n(z,q)/C_n(q)$ and ${\mathrm{ge}}_n(z,q)/S_n(q)$ are invariant.) … ►For further information on $C_n(q)$, $S_n(q)$, and expansions of $f_n(z,q)$, $g_n(z,q)$ in Fourier series or in series of ${\mathrm{ce}}_n$, ${\mathrm{se}}_n$ functions, see McLachlan (1947, Chapter VII) or Meixner and Schäfke (1954, §2.72). …

4: 31.8 Solutions via Quadratures

… ►the Hermite–Darboux method (see Whittaker and Watson (1927, pp. 570–572)) can be applied to construct solutions of (31.2.1) expressed in quadratures, as follows. … ►Here ${\mathrm{\Psi }}_{g,N}(\lambda ,z)$ is a polynomial of degree $g$ in $\lambda$ and of degree $N=m_0+m_1+m_2+m_3$ in $z$, that is a solution of the third-order differential equation satisfied by a product of any two solutions of Heun’s equation. …(This $\nu$ is unrelated to the $\nu$ in §31.6.) … ►By automorphisms from §31.2(v), similar solutions also exist for $m_0,m_1,m_2,m_3\in \mathbb{Z}$, and ${\mathrm{\Psi }}_{g,N}(\lambda ,z)$ may become a rational function in $z$. …The curve $\mathrm{\Gamma }$ reflects the finite-gap property of Equation (31.2.1) when the exponent parameters satisfy (31.8.1) for $m_j\in \mathbb{Z}$. …

5: 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). …

6: 28.31 Equations of Whittaker–Hill and Ince

… ►Hill’s equation with three terms … ►in (28.31.1). … ►and $m=0,1,\mathrm{\dots },n$ in all cases. … ►They are real and distinct, and can be ordered so that $C_p^m(z,\xi )$ and $S_p^m(z,\xi )$ have precisely $m$ zeros, all simple, in . … ►More important are the double orthogonality relations for $p_1\ne p_2$ or $m_1\ne m_2$ or both, given by …

7: 4.37 Inverse Hyperbolic Functions

… ►Elsewhere on the integration paths in (4.37.1) and (4.37.2) the branches are determined by continuity. … ►The principal values (or principal branches) of the inverse $\sinh$, $\cosh$, and $\tanh$ are obtained by introducing cuts in the $z$-plane as indicated in Figure 4.37.1(i)-(iii), and requiring the integration paths in (4.37.1)–(4.37.3) not to cross these cuts. … ►These functions are analytic in the cut plane depicted in Figure 4.37.1(iv), (v), (vi), respectively. … ►

§4.37(iii) Reflection Formulas

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§4.37(v) Fundamental Property

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8: 4.23 Inverse Trigonometric Functions

… ►The principal values (or principal branches) of the inverse sine, cosine, and tangent are obtained by introducing cuts in the $z$-plane as indicated in Figures 4.23.1(i) and 4.23.1(ii), and requiring the integration paths in (4.23.1)–(4.23.3) not to cross these cuts. … ►These functions are analytic in the cut plane depicted in Figures 4.23.1(iii) and 4.23.1(iv). … ►

§4.23(iii) Reflection Formulas

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§4.23(v) Fundamental Property

… ►where $z=x+\mathrm{i}y$ and $\pm z\notin (1,\mathrm{\infty })$ in (4.23.34) and (4.23.35), and in (4.23.36). …

9: 10.47 Definitions and Basic Properties

§10.47 Definitions and Basic Properties

… ►Many properties of ${𝗃}_n\left(z\right)$, ${𝗒}_n\left(z\right)$, ${𝗁}_n^{(1)}\left(z\right)$, ${𝗁}_n^{(2)}\left(z\right)$, ${𝗂}_n^{(1)}\left(z\right)$, ${𝗂}_n^{(2)}\left(z\right)$, and ${𝗄}_n\left(z\right)$ follow straightforwardly from the above definitions and results given in preceding sections of this chapter. … … ►For (10.47.2) numerically satisfactory pairs of solutions are ${𝗂}_n^{(1)}\left(z\right)$ and ${𝗄}_n\left(z\right)$ in the right half of the $z$-plane, and ${𝗂}_n^{(1)}\left(z\right)$ and ${𝗄}_n\left(-z\right)$ in the left half of the $z$-plane. … ►

§10.47(v) Reflection Formulas

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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

… ►In particular, … ►

§10.61(iv) Reflection Formulas for Orders

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