relations to trigonometric functions

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21: 7.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) … ►Unless otherwise noted, primes indicate derivatives with respect to the argument. ►The main functions treated in this chapter are the error function

\operatorname{erf}z

; the complementary error functions

\operatorname{erfc}z

and

w\left(z\right)

; Dawson’s integral

F\left(z\right)

; the Fresnel integrals

\mathcal{F}\left(z\right)

C\left(z\right)

, and

S\left(z\right)

; the Goodwin–Staton integral

G\left(z\right)

; the repeated integrals of the complementary error function

\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)

; the Voigt functions

\mathsf{U}\left(x,t\right)

and

\mathsf{V}\left(x,t\right)

. ►Alternative notations are

Q(z)=\tfrac{1}{2}\operatorname{erfc}\left(z/\sqrt{2}\right)

P(z)=\Phi(z)=\tfrac{1}{2}\operatorname{erfc}\left(-z/\sqrt{2}\right)

\operatorname{Erf}z=\tfrac{1}{2}\sqrt{\pi}\operatorname{erf}z

\operatorname{Erfi}z=e^{z^{2}}F\left(z\right)

C_{1}(z)=C\left(\sqrt{2/\pi}z\right)

S_{1}(z)=S\left(\sqrt{2/\pi}z\right)

C_{2}(z)=C\left(\sqrt{2z/\pi}\right)

S_{2}(z)=S\left(\sqrt{2z/\pi}\right)

. … ►

22: 10.16 Relations to Other Functions

§10.16 Relations to Other Functions

►

Elementary Functions

… ►

Parabolic Cylinder Functions

… ►

Confluent Hypergeometric Functions

… ►

Generalized Hypergeometric Functions

…

23: 14.19 Toroidal (or Ring) Functions

§14.19 Toroidal (or Ring) Functions

►

§14.19(i) Introduction

… ►This form of the differential equation arises when Laplace’s equation is transformed into toroidal coordinates

(\eta,\theta,\phi)

, which are related to Cartesian coordinates

(x,y,z)

by … ►

§14.19(iv) Sums

… ►

§14.19(v) Whipple’s Formula for Toroidal Functions

…

24: 14.3 Definitions and Hypergeometric Representations

§14.3 Definitions and Hypergeometric Representations

►

§14.3(i) Interval $-1<x<1$

… ►

§14.3(ii) Interval $1<x<\infty$

… ►

§14.3(iii) Alternative Hypergeometric Representations

… ►

§14.3(iv) Relations to Other Functions

…

25: 14.5 Special Values

§14.5 Special Values

… ►

§14.5(v) $\mu=0$ , $\nu=\pm\frac{1}{2}$

… ► … ►

14.5.26

\boldsymbol{Q}_{\frac{1}{2}}\left(\cosh\xi\right)=2\pi^{-1/2}\cosh\xi% \operatorname{sech}\left(\tfrac{1}{2}\xi\right)K\left(\operatorname{sech}\left% (\tfrac{1}{2}\xi\right)\right)-4\pi^{-1/2}\cosh\left(\tfrac{1}{2}\xi\right)E% \left(\operatorname{sech}\left(\tfrac{1}{2}\xi\right)\right),

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $\cosh\NVar{z}$ : hyperbolic cosine function, $\operatorname{sech}\NVar{z}$ : hyperbolic secant function, $\boldsymbol{Q}_{\NVar{\nu}}\left(\NVar{z}\right)=\boldsymbol{Q}^{0}_{\nu}\left% (z\right)$ : Olver’s associated Legendre function and $\xi>0$ : variable
A&S Ref:: 8.13.7 (in slightly different form)
Permalink:: http://dlmf.nist.gov/14.5.E26
Encodings:: pMML, png, TeX
See also:: Annotations for §14.5(v), §14.5 and Ch.14

… ►

§14.5(vi) Addendum to §14.5(ii): $\mu=0$ , $\nu=2$

…

26: 34.3 Basic Properties: $\mathit{3j}$ Symbol

… ►

§34.3(iii) Recursion Relations

… ►For these and other recursion relations see Varshalovich et al. (1988, §8.6). … ►

§34.3(vii) Relations to Legendre Polynomials and Spherical Harmonics

… ►Equations (34.3.19)–(34.3.22) are particular cases of more general results that relate rotation matrices to

\mathit{3j}

symbols, for which see Edmonds (1974, Chapter 4). …

27: 30.14 Wave Equation in Oblate Spheroidal Coordinates

… ►Oblate spheroidal coordinates

\xi,\eta,\phi

are related to Cartesian coordinates

x, y, z

by … ►The wave equation (30.13.7), transformed to oblate spheroidal coordinates

(\xi,\eta,\phi)

, admits solutions of the form (30.13.8), where

w_{1}

satisfies the differential equation …Equation (30.14.7) can be transformed to equation (30.2.1) by the substitution

z=\pm i\xi

. ►In most applications the solution

w

has to be a single-valued function of

(x,y,z)

, which requires

\mu=m

(a nonnegative integer). Moreover, the solution

w

has to be bounded along the

z

-axis: this requires

w_{2}(\eta)

to be bounded when

-1<\eta<1

. …

28: 30.13 Wave Equation in Prolate Spheroidal Coordinates

… ►Prolate spheroidal coordinates

\xi,\eta,\phi

are related to Cartesian coordinates

x, y, z

by …The

(x,y,z)

-space without the

z

-axis corresponds to … ►transformed to prolate spheroidal coordinates

(\xi,\eta,\phi)

, admits solutions …where

w_{1}

w_{2}

w_{3}

satisfy the differential equations … ►In most applications the solution

w

has to be a single-valued function of

(x,y,z)

, which requires

\mu=m

(a nonnegative integer) and …

29: 29.6 Fourier Series

§29.6 Fourier Series

►

§29.6(i) Function $\mathit{Ec}^{2m}_{\nu}\left(z,k^{2}\right)$

… ►In addition, if

H

satisfies (29.6.2), then (29.6.3) applies. … ►Consequently,

\mathit{Ec}^{2m}_{\nu}\left(z,k^{2}\right)

reduces to a Lamé polynomial; compare §§29.12(i) and 29.15(i). … ►

§29.6(ii) Function $\mathit{Ec}^{2m+1}_{\nu}\left(z,k^{2}\right)$

…

30: 36.7 Zeros

… ►Close to the

y

-axis the approximate location of these zeros is given by … ►Deep inside the bifurcation set, that is, inside the three-cusped astroid (36.4.10) and close to the part of the

z

-axis that is far from the origin, the zero contours form an array of rings close to the planes …,

y=0

), the number of rings in the

m

th row, measured from the origin and before the transition to hairpins, is given by …Outside the bifurcation set (36.4.10), each rib is flanked by a series of zero lines in the form of curly “antelope horns” related to the “outside” zeros (36.7.2) of the cusp canonical integral. There are also three sets of zero lines in the plane

z=0

related by

2\pi/3

rotation; these are zeros of (36.2.20), whose asymptotic form in polar coordinates

(x=r\cos\theta,\;y=r\sin\theta)

is given by …