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31: 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).

32: Sidebar 7.SB1: Diffraction from a Straightedge

… ►The intensity distribution follows ${|ℱ\left(x\right)|}^2$, where $ℱ$ is the Fresnel integral (See 7.3.4). Fresnel integrals have many applications in optics. …

33: 36.14 Other Physical Applications

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§36.14(i) Caustics

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§36.14(ii) Optics

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§36.14(iii) Quantum Mechanics

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§36.14(iv) Acoustics

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34: 19.6 Special Cases

§19.6 Special Cases

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§19.6(i) Complete Elliptic Integrals

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§19.6(ii) $F(\varphi ,k)$

… ►Circular and hyperbolic cases, including Cauchy principal values, are unified by using $R_C(x,y)$. … ►

§19.6(v) $R_C(x,y)$

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35: 19.10 Relations to Other Functions

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§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

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$$\ln \left(x/y\right)=(x-y)R_C(\frac{1}{4}{(x+y)}^2,xy),$$

… ►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. …

36: 6.21 Software

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§6.21(ii) $E_1\left(x\right)$, $\mathrm{Ei}\left(x\right)$, $\mathrm{Si}\left(x\right)$, $\mathrm{Ci}\left(x\right)$, $\mathrm{Shi}\left(x\right)$, $\mathrm{Chi}\left(x\right)$, $x\in ℝ$

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§6.21(iii) $E_1\left(z\right)$, $\mathrm{Si}\left(z\right)$, $\mathrm{Ci}\left(z\right)$, $\mathrm{Shi}\left(z\right)$, $\mathrm{Chi}\left(z\right)$, $z\in ℂ$

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37: 7.5 Interrelations

§7.5 Interrelations

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7.5.2 $$C\left(z\right)+\mathrm{i}S\left(z\right)=\frac{1}{2}(1+\mathrm{i})-ℱ\left(z\right).$$

ⓘ

Symbols:

$C\left(z\right)$: Fresnel integral, $ℱ\left(z\right)$: Fresnel integral, $S\left(z\right)$: 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 $\mathrm{Ei}\left(x\right)$ see §6.2(i).

38: 8.24 Physical Applications

§8.24 Physical Applications

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§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)).

39: 19.24 Inequalities

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§19.24(i) Complete Integrals

… ►For $x>0$, $y>0$, and $x\ne y$, the complete cases of $R_F$ and $R_G$ satisfy … ►

§19.24(ii) Incomplete Integrals

… ►Other inequalities for $R_F(x,y,z)$ are given in Carlson (1970). …

40: 19.27 Asymptotic Approximations and Expansions

§19.27 Asymptotic Approximations and Expansions

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§19.27(ii) $R_F(x,y,z)$

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§19.27(iii) $R_G(x,y,z)$

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§19.27(iv) $R_D(x,y,z)$

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