relation to logarithmic integral

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11: 7.10 Derivatives

§7.10 Derivatives

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7.10.1

\frac{{\mathrm{d}}^{n+1}\operatorname{erf}z}{{\mathrm{d}z}^{n+1}}=(-1)^{n}% \frac{2}{\sqrt{\pi}}H_{n}\left(z\right)e^{-z^{2}},

n=0,1,2,\dots

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Symbols:: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\operatorname{erf}\NVar{z}$ : 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'\left(z\right)=-2zw\left(z\right)+(2i/\sqrt{\pi}),

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $w\left(\NVar{z}\right)$ : 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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\frac{\mathrm{d}\mathrm{g}\left(z\right)}{\mathrm{d}z}=\pi z\mathrm{f}\left(z% \right)-1.

12: 7.11 Relations to Other Functions

§7.11 Relations to Other Functions

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Incomplete Gamma Functions and Generalized Exponential Integral

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Confluent Hypergeometric Functions

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7.11.6

C\left(z\right)+iS\left(z\right)=zM\left(\tfrac{1}{2},\tfrac{3}{2},\tfrac{1}{2% }\pi iz^{2}\right)=ze^{\pi iz^{2}/2}M\left(1,\tfrac{3}{2},-\tfrac{1}{2}\pi iz^% {2}\right).

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Symbols:: $C\left(\NVar{z}\right)$ : Fresnel integral, $S\left(\NVar{z}\right)$ : Fresnel integral, $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/7.11.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §7.11, §7.11 and Ch.7

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Generalized Hypergeometric Functions

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13: 7.19 Voigt Functions

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7.19.2

\mathsf{V}\left(x,t\right)=\frac{1}{\sqrt{4\pi t}}\int_{-\infty}^{\infty}\frac% {ye^{-(x-y)^{2}/(4t)}}{1+y^{2}}\,\mathrm{d}y.

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Defines:: $\mathsf{V}\left(\NVar{x},\NVar{t}\right)$ : Voigt function
Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : 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\left(a,u\right)=\frac{a}{\pi}\int_{-\infty}^{\infty}\frac{e^{-t^{2}}\,% \mathrm{d}t}{(u-t)^{2}+a^{2}}=\frac{1}{a\sqrt{\pi}}\mathsf{U}\left(\frac{u}{a}% ,\frac{1}{4a^{2}}\right).

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Defines:: $H\left(\NVar{a},\NVar{u}\right)$ : line-broadening function
Symbols:: $\mathsf{U}\left(\NVar{x},\NVar{t}\right)$ : Voigt function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : 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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14: 19.14 Reduction of General Elliptic Integrals

§19.14 Reduction of General Elliptic Integrals

… ►Legendre (1825–1832) showed that every elliptic integral can be expressed in terms of the three integrals in (19.1.2) supplemented by algebraic, logarithmic, and trigonometric functions. The classical method of reducing (19.2.3) to Legendre’s integrals is described in many places, especially Erdélyi et al. (1953b, §13.5), Abramowitz and Stegun (1964, Chapter 17), and Labahn and Mutrie (1997, §3). …A similar remark applies to the transformations given in Erdélyi et al. (1953b, §13.5) and to the choice among explicit reductions in the extensive table of Byrd and Friedman (1971), in which one limit of integration is assumed to be a branch point of the integrand at which the integral converges. …

15: 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: … ►The

A_{k}\left(z,p\right)

are related by … ►Further properties of these functions, and also of similar contour integrals containing an additional factor

(\ln t)^{q}

q=1,2,\ldots

, in the integrand, are derived in Reid (1972), Drazin and Reid (1981, Appendix), and Baldwin (1985). …

16: 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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17: 7.13 Zeros

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§7.13(iii) Zeros of the Fresnel Integrals

… ►As

n\to\infty

the

x_{n}

and

y_{n}

corresponding to the zeros of

C\left(z\right)

satisfy … ►As

n\to\infty

the

x_{n}

and

y_{n}

corresponding to the zeros of

S\left(z\right)

satisfy (7.13.5) with … ►In consequence of (7.5.5) and (7.5.10), zeros of

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

are related to zeros of

\operatorname{erfc}z

. … ►

18: 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\infty

… ►(1.8.10) continues to apply if either

a

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}\mathscr{F}\left(f\right)\left(-2\pi n\right)

. …

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: 19.25 Relations to Other Functions

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§19.25(ii) Bulirsch’s Integrals as Symmetric Integrals

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§19.25(iv) Theta Functions

… ► … ►

§19.25(vii) Hypergeometric Function

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