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11: 14.7 Integer Degree and Order

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§14.7(iii) Reflection Formulas

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12: 19.3 Graphics

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Figure 19.3.9: $\mathrm{\Re }\left(K\left(k\right)\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. The real part is symmetric under reflection in the real axis. … Magnify 3D Help

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Figure 19.3.10: $\mathrm{\Im }\left(K\left(k\right)\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. The imaginary part is 0 for , and is antisymmetric under reflection in the real axis. … Magnify 3D Help

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Figure 19.3.11: $\mathrm{\Re }\left(E\left(k\right)\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. The real part is symmetric under reflection in the real axis. … Magnify 3D Help

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Figure 19.3.12: $\mathrm{\Im }\left(E\left(k\right)\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. The imaginary part is 0 for $k^2\le 1$ and is antisymmetric under reflection in the real axis. … Magnify 3D Help

13: 11.9 Lommel Functions

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Reflection Formulas

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14: 1.10 Functions of a Complex Variable

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Schwarz Reflection Principle

… ►Then $f(z)$ can be continued analytically across $\mathrm{𝐴𝐵}$ by reflection, that is, …

15: 18.38 Mathematical Applications

… ►The Dunkl operator, introduced by Dunkl (1989), is an operator associated with reflection groups or root systems which has terms involving first order partial derivatives and reflection terms. … ►The Dunkl type operator is a $q$-difference-reflection operator acting on Laurent polynomials and its eigenfunctions, the nonsymmetric Askey–Wilson polynomials, are linear combinations of the symmetric Laurent polynomial $R_n(z;a,b,c,d|q)$ and the ‘anti-symmetric’ Laurent polynomial $z^{-1}(1-az)(1-bz)R_{n-1}(z;qa,qb,c,d|q)$, where $R_n(z)$ is given in (18.28.1_5). …

16: 10.47 Definitions and Basic Properties

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§10.47(v) Reflection Formulas

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17: 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 $m_j\in \mathbb{Z}$. …

18: Bibliography F

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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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Referenced by:

§36.14(iv).

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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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Referenced by:

§36.14(iv).

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19: 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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20: 35.7 Gaussian Hypergeometric Function of Matrix Argument

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Reflection Formula

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