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1: 25.4 Reflection Formulas

§25.4 Reflection Formulas

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25.4.1 $$\zeta \left(1-s\right)=2{(2\pi )}^{-s}\cos \left(\frac{1}{2}\pi s\right)\mathrm{\Gamma }\left(s\right)\zeta \left(s\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function and $s$: complex variable

Keywords:

reflection formula

Source:

Apostol (1976, (12), p. 259)

Referenced by:

§25.10(i), §5.17

Permalink:

http://dlmf.nist.gov/25.4.E1

Encodings:

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See also:

Annotations for §25.4 and Ch.25

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25.4.2 $$\zeta \left(s\right)=2{(2\pi )}^{s-1}\sin \left(\frac{1}{2}\pi s\right)\mathrm{\Gamma }\left(1-s\right)\zeta \left(1-s\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function and $s$: complex variable

Keywords:

reflection formula

Source:

Apostol (1976, (13), p. 259)

A&S Ref:

23.2.6

Permalink:

http://dlmf.nist.gov/25.4.E2

Encodings:

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See also:

Annotations for §25.4 and Ch.25

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25.4.3 $$\xi \left(s\right)=\xi \left(1-s\right),$$

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Symbols:

$\xi \left(s\right)$: Riemann’s $\xi$-function and $s$: complex variable

Keywords:

reflection formula

Source:

Apostol (1976, p. 260)

Permalink:

http://dlmf.nist.gov/25.4.E3

Encodings:

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See also:

Annotations for §25.4 and Ch.25

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2: 5.21 Methods of Computation

… ►For the left half-plane we can continue the backward recurrence or make use of the reflection formula (5.5.3). …

3: Preface

… ►The term digital library has gained acceptance for this kind of information resource, and our choice of project title reflects our hope that the NIST DLMF will be a vehicle for revolutionizing the way applicable mathematics in general is practiced and delivered.

4: 5.5 Functional Relations

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§5.5(ii) Reflection

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5.5.3 $$\mathrm{\Gamma }\left(z\right)\mathrm{\Gamma }\left(1-z\right)=\pi /\sin \left(\pi z\right),$$ $z\ne 0,\pm 1,\mathrm{\dots }$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function and $z$: complex variable

A&S Ref:

6.1.17 (without the condition on $z$.)

Referenced by:

§10.19(i), §15.4(ii), §15.8(ii), (25.9.2), §5.21, (9.10.18), (9.12.15)

Permalink:

http://dlmf.nist.gov/5.5.E3

Encodings:

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See also:

Annotations for §5.5(ii), §5.5 and Ch.5

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5: 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:

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See also:

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

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6: 37.19 Other Orthogonal Polynomials of $d$ Variables

… ►These are orthogonal polynomials for a family of reflection invariant weight functions on the unit sphere. Let $G$ be a reflection group (also called Coxeter group) with reduced root system $R$ (see Dunkl and Xu (2014, §6.2) for definitions of these notions). …The weight function is invariant under the reflection group $G$. Often the notations $h_{\kappa }^2(𝐱)$ and ${\mathrm{\Delta }}_h$ are used instead of $w_{\kappa }(𝐱)$ and ${\mathrm{\Delta }}_{\kappa }$, respectively, which explains the term $h$-harmonics. … ►Let $w_{\kappa }$ be the reflection invariant weight function defined in (37.19.4). …

7: 36.14 Other Physical Applications

… ►Applications include the reflection of ultrasound pulses, and acoustical waveguides. …

8: 10.61 Definitions and Basic Properties

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§10.61(iii) Reflection Formulas for Arguments

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§10.61(iv) Reflection Formulas for Orders

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9: 34.1 Special Notation

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34.1.1

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Symbols:

: $\mathit{3}j$ symbol and $j,j_r$: non-negative integers or non-negative integers plus one half.

Sources:

Edmonds (1974, p. 46, Eq. (3.7.3)); Rotenberg et al. (1959, p. 1, Eq. (1.1a))

Permalink:

http://dlmf.nist.gov/34.1.E1

Encodings:

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Clarification (effective with 1.0.15):

A wording change reflects that the Clebsch–Gordan coefficients are an alternative formulation of angular momentum problems, rather than alternative notation for the $\mathit{3}j$. The sign of $m_3$ was changed for clarity.

See also:

Annotations for §34.1 and Ch.34

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10: 4.37 Inverse Hyperbolic Functions

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§4.37(iii) Reflection Formulas

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