Basset integral

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31: 6.7 Integral Representations

§6.7 Integral Representations

►

§6.7(i) Exponential Integrals

… ► ►

§6.7(ii) Sine and Cosine Integrals

… ►

§6.7(iii) Auxiliary Functions

…

32: Sidebar 7.SB1: Diffraction from a Straightedge

… ►The intensity distribution follows

|\mathcal{F}\left(x\right)|^{2}

, where

\mathcal{F}

is the Fresnel integral (See 7.3.4). Fresnel integrals have many applications in optics. …

33: 12.16 Mathematical Applications

… ►PCFs are used as basic approximating functions in the theory of contour integrals with a coalescing saddle point and an algebraic singularity, and in the theory of differential equations with two coalescing turning points; see §§2.4(vi) and 2.8(vi). … ► ►PCFs are also used in integral transforms with respect to the parameter, and inversion formulas exist for kernels containing PCFs. …Integral transforms and sampling expansions are considered in Jerri (1982).

34: 36.14 Other Physical Applications

… ►

§36.14(i) Caustics

… ►

§36.14(ii) Optics

… ►

§36.14(iii) Quantum Mechanics

… ►

§36.14(iv) Acoustics

…

35: 19.6 Special Cases

§19.6 Special Cases

►

§19.6(i) Complete Elliptic Integrals

… ►

§19.6(ii) $F\left(\phi,k\right)$

… ►Circular and hyperbolic cases, including Cauchy principal values, are unified by using

R_{C}\left(x,y\right)

. … ►

§19.6(v) $R_{C}\left(x,y\right)$

…

36: 19.10 Relations to Other Functions

… ►

§19.10(i) Theta and Elliptic Functions

►For relations of Legendre’s integrals to theta functions, Jacobian functions, and Weierstrass functions, see §§20.9(i), 22.15(ii), and 23.6(iv), respectively. … ►

§19.10(ii) Elementary Functions

… ►

\ln\left(x/y\right)=(x-y)R_{C}\left(\tfrac{1}{4}(x+y)^{2},xy\right),

… ►In each case when

y=1

, the quantity multiplying

R_{C}

supplies the asymptotic behavior of the left-hand side as the left-hand side tends to 0. …

37: 6.21 Software

… ►

§6.21(ii) $E_{1}\left(x\right)$ , $\operatorname{Ei}\left(x\right)$ , $\operatorname{Si}\left(x\right)$ , $\operatorname{Ci}\left(x\right)$ , $\operatorname{Shi}\left(x\right)$ , $\operatorname{Chi}\left(x\right)$ , $x\in\mathbb{R}$

… ►

§6.21(iii) $E_{1}\left(z\right)$ , $\operatorname{Si}\left(z\right)$ , $\operatorname{Ci}\left(z\right)$ , $\operatorname{Shi}\left(z\right)$ , $\operatorname{Chi}\left(z\right)$ , $z\in\mathbb{C}$

…

38: 7.5 Interrelations

§7.5 Interrelations

… ►

7.5.2

C\left(z\right)+iS\left(z\right)=\tfrac{1}{2}(1+i)-\mathcal{F}\left(z\right).

ⓘ

Symbols:: $C\left(\NVar{z}\right)$ : Fresnel integral, $\mathcal{F}\left(\NVar{z}\right)$ : Fresnel integral, $S\left(\NVar{z}\right)$ : Fresnel integral, $\mathrm{i}$ : imaginary unit and $z$ : complex variable
Referenced by:: §7.5
Permalink:: http://dlmf.nist.gov/7.5.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §7.5 and Ch.7

… ► … …For

\operatorname{Ei}\left(x\right)

see §6.2(i).

39: 8.24 Physical Applications

§8.24 Physical Applications

… ►

§8.24(iii) Generalized Exponential Integral

►The function

E_{p}\left(x\right)

, with

p>0

, appears in theories of transport and radiative equilibrium (Hopf (1934), Kourganoff (1952), Altaç (1996)). ►With more general values of

p

E_{p}\left(x\right)

supplies fundamental auxiliary functions that are used in the computation of molecular electronic integrals in quantum chemistry (Harris (2002), Shavitt (1963)), and also wave acoustics of overlapping sound beams (Ding (2000)).

40: 19.24 Inequalities

… ►

§19.24(i) Complete Integrals

… ►For

x>0

y>0

, and

x\neq y

, the complete cases of

R_{F}

and

R_{G}

satisfy … ►

§19.24(ii) Incomplete Integrals

… ►Other inequalities for

R_{F}\left(x,y,z\right)

are given in Carlson (1970). …