relations to exponential integrals

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11: 36.2 Catastrophes and Canonical Integrals

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Canonical Integrals

… ► ${\mathrm{\Psi }}_1$ is related to the Airy function (§9.2): … … ►Addendum: For further special cases see §36.2(iv) … ►

§36.2(iv) Addendum to 36.2(ii) Special Cases

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12: 7.18 Repeated Integrals of the Complementary Error Function

§7.18 Repeated Integrals of the Complementary Error Function

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Hermite Polynomials

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Confluent Hypergeometric Functions

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Parabolic Cylinder Functions

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Probability Functions

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13: 12.7 Relations to Other Functions

§12.7 Relations to Other Functions

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§12.7(i) Hermite Polynomials

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§12.7(ii) Error Functions, Dawson’s Integral, and Probability Function

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§12.7(iii) Modified Bessel Functions

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§12.7(iv) Confluent Hypergeometric Functions

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14: 1.8 Fourier Series

… ►Here $c_n$ is related to $a_n$ and $b_n$ in (1.8.1), (1.8.2) by $c_n=\frac{1}{2}(a_n-\mathrm{i}{b}_n)$, $c_{-n}=\frac{1}{2}(a_n+\mathrm{i}{b}_n)$ for $n>0$ and $c_0=\frac{1}{2}{a}_0$. … ►As $n\to \mathrm{\infty }$ … ►(1.8.10) continues to apply if either $a$ or $b$ or both are infinite and/or $f(x)$ has finitely many singularities in $(a,b)$, provided that the integral converges uniformly (§1.5(iv)) at $a,b$, and the singularities for all sufficiently large $\lambda$. … ►Then the series (1.8.1) converges to the sum … ►It follows from definition (1.14.1) that the integral in (1.8.14) is equal to $\sqrt{2\pi }ℱ\left(f\right)\left(-2\pi n\right)$. …

15: 20.9 Relations to Other Functions

§20.9 Relations to Other Functions

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§20.9(i) Elliptic Integrals

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§20.9(ii) Elliptic Functions and Modular Functions

►See §§22.2 and 23.6(i) for the relations of Jacobian and Weierstrass elliptic functions to theta functions. … ►

§20.9(iii) Riemann Zeta Function

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16: 7.19 Voigt Functions

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7.19.2 $$𝖵(x,t)=\frac{1}{\sqrt{4\pi t}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{y{\mathrm{e}}^{-{(x-y)}^2/(4t)}}{1+y^2}dy.$$

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Defines:

$𝖵(x,t)$: Voigt function

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $x$: real variable

Referenced by:

§7.19(iii)

Permalink:

http://dlmf.nist.gov/7.19.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §7.19(i), §7.19 and Ch.7

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7.19.4 $$H(a,u)=\frac{a}{\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-t^2}dt}{{(u-t)}^2+a^2}=\frac{1}{a\sqrt{\pi }}𝖴(\frac{u}{a},\frac{1}{4a^2}).$$

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Defines:

$H(a,u)$: line-broadening function

Symbols:

$𝖴(x,t)$: Voigt function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm and $\int$: integral

Permalink:

http://dlmf.nist.gov/7.19.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §7.19(i), §7.19 and Ch.7

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17: 7.10 Derivatives

§7.10 Derivatives

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7.10.1 $$\frac{d^{n+1}\mathrm{erf}z}{{dz}^{n+1}}={(-1)}^n\frac{2}{\sqrt{\pi }}{H}_n\left(z\right){\mathrm{e}}^{-z^2},$$ $n=0,1,2,\mathrm{\dots }$.

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.1.19

Permalink:

http://dlmf.nist.gov/7.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §7.10 and Ch.7

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7.10.2 $$w^{\prime }\left(z\right)=-2zw\left(z\right)+(2\mathrm{i}/\sqrt{\pi }),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $w\left(z\right)$: Faddeeva (or Faddeyeva) function, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

7.1.20

Permalink:

http://dlmf.nist.gov/7.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §7.10 and Ch.7

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${\displaystyle \frac{d\mathrm{g}\left(z\right)}{dz}}=\pi z\mathrm{f}\left(z\right)-1.$

18: 13.18 Relations to Other Functions

§13.18 Relations to Other Functions

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§13.18(i) Elementary Functions

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§13.18(iv) Parabolic Cylinder Functions

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§13.18(v) Orthogonal Polynomials

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Laguerre Polynomials

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19: 13.6 Relations to Other Functions

§13.6 Relations to Other Functions

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§13.6(iv) Parabolic Cylinder Functions

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§13.6(v) Orthogonal Polynomials

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Laguerre Polynomials

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§13.6(vi) Generalized Hypergeometric Functions

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20: 9.13 Generalized Airy Functions

… ►Swanson and Headley (1967) define independent solutions $A_n\left(z\right)$ and $B_n\left(z\right)$ of (9.13.1) by … ►Their relations to the functions $A_n\left(z\right)$ and $B_n\left(z\right)$ are given by … ►When $\alpha$ is a positive integer the relation of these functions to $W_m(t)$, $W_m(-t)$ is as follows: … ►Reid (1972) and Drazin and Reid (1981, Appendix) introduce the following contour integrals in constructing approximate solutions to the Orr–Sommerfeld equation for fluid flow: … ►The $A_k(z,p)$ are related by …