repeated%20integrals%20of%20the%20complementary%20error%20function

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1: 7.18 Repeated Integrals of the Complementary Error Function

§7.18 Repeated Integrals of the Complementary Error Function

►

§7.18(i) Definition

… ►

§7.18(iii) Properties

… ► … ►

Hermite Polynomials

…

2: 1.14 Integral Transforms

§1.14 Integral Transforms

… ►where the last integral denotes the Cauchy principal value (1.4.25). … ►If

x^{\sigma-1}f(x)

is integrable on

(0,\infty)

for all

\sigma

a<\sigma<b

, then the integral (1.14.32) converges and

\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(s\right)

is an analytic function of

s

in the vertical strip

a<\Re s<b

. … ►

§1.14(viii) Compendia

►For more extensive tables of the integral transforms of this section and tables of other integral transforms, see Erdélyi et al. (1954a, b), Gradshteyn and Ryzhik (2000), Marichev (1983), Oberhettinger (1972, 1974, 1990), Oberhettinger and Badii (1973), Oberhettinger and Higgins (1961), Prudnikov et al. (1986a, b, 1990, 1992a, 1992b).

3: 7.2 Definitions

… ►

§7.2(i) Error Functions

… ►

\operatorname{erf}z

\operatorname{erfc}z

, and

w\left(z\right)

are entire functions of

z

, as is

F\left(z\right)

in the next subsection. … ►

§7.2(ii) Dawson’s Integral

… ►

§7.2(iii) Fresnel Integrals

… ►

§7.2(iv) Auxiliary Functions

…

4: 8.19 Generalized Exponential Integral

§8.19 Generalized Exponential Integral

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§8.19(i) Definition and Integral Representations

… ►

Other Integral Representations

… ►

§8.19(vi) Relation to Confluent Hypergeometric Function

… ►

§8.19(x) Integrals

…

5: 6.2 Definitions and Interrelations

… ►

§6.2(i) Exponential and Logarithmic Integrals

… ►

\operatorname{Ein}\left(z\right)

is sometimes called the complementary exponential integral. … ►The logarithmic integral is defined by … ►

§6.2(ii) Sine and Cosine Integrals

… ►

§6.2(iii) Auxiliary Functions

…

6: 19.16 Definitions

… ►

§19.16(i) Symmetric Integrals

… ►Just as the elementary function

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

(§19.2(iv)) is the degenerate case … ►

§19.16(ii) $R_{-a}\left(\mathbf{b};\mathbf{z}\right)$

►All elliptic integrals of the form (19.2.3) and many multiple integrals, including (19.23.6) and (19.23.6_5), are special cases of a multivariate hypergeometric function …The

R

-function is often used to make a unified statement of a property of several elliptic integrals. …

7: 8.21 Generalized Sine and Cosine Integrals

§8.21 Generalized Sine and Cosine Integrals

… ►

§8.21(iii) Integral Representations

… ►

Spherical-Bessel-Function Expansions

… ►

§8.21(viii) Asymptotic Expansions

… ►

8: 19.2 Definitions

… ►

§19.2(i) General Elliptic Integrals

… ►

§19.2(ii) Legendre’s Integrals

… ►Legendre’s complementary complete elliptic integrals are defined via … ►

§19.2(iii) Bulirsch’s Integrals

… ►

§19.2(iv) A Related Function: $R_{C}\left(x,y\right)$

…

9: 36.2 Catastrophes and Canonical Integrals

§36.2 Catastrophes and Canonical Integrals

… ►

Canonical Integrals

… ►

\Psi_{1}

is related to the Airy function (§9.2): … … ►

§36.2(iii) Symmetries

…

10: 7.23 Tables

… ►

•

Abramowitz and Stegun (1964, Chapter 7) includes $\operatorname{erf}x$ , $(2/\sqrt{\pi})e^{-x^{2}}$ , $x\in[0,2]$ , 10D; $(2/\sqrt{\pi})e^{-x^{2}}$ , $x\in[2,10]$ , 8S; $xe^{x^{2}}\operatorname{erfc}x$ , $x^{-2}\in[0,0.25]$ , 7D; $2^{n}\Gamma\left(\frac{1}{2}n+1\right)\mathop{\mathrm{i}^{n}\mathrm{erfc}}% \left(x\right)$ , $n=1(1)6,10,11$ , $x\in[0,5]$ , 6S; $F\left(x\right)$ , $x\in[0,2]$ , 10D; $xF\left(x\right)$ , $x^{-2}\in[0,0.25]$ , 9D; $C\left(x\right)$ , $S\left(x\right)$ , $x\in[0,5]$ , 7D; $\mathrm{f}\left(x\right)$ , $\mathrm{g}\left(x\right)$ , $x\in[0,1]$ , $x^{-1}\in[0,1]$ , 15D.

►

•

Abramowitz and Stegun (1964, Table 27.6) includes the Goodwin–Staton integral $G\left(x\right)$ , $x=1(.1)3(.5)8$ , 4D; also $G\left(x\right)+\ln x$ , $x=0(.05)1$ , 4D.

… ►

•

Zhang and Jin (1996, pp. 637, 639) includes $(2/\sqrt{\pi})e^{-x^{2}}$ , $\operatorname{erf}x$ , $x=0(.02)1(.04)3$ , 8D; $C\left(x\right)$ , $S\left(x\right)$ , $x=0(.2)10(2)100(100)500$ , 8D.

… ►

•

Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of $\operatorname{erf}z$ , $x\in[0,5]$ , $y=0.5(.5)3$ , 7D and 8D, respectively; the real and imaginary parts of $\int_{x}^{\infty}e^{\pm\mathrm{i}t^{2}}\,\mathrm{d}t$ , $(1/\sqrt{\pi})e^{\mp\mathrm{i}(x^{2}+(\pi/4))}\int_{x}^{\infty}e^{\pm\mathrm{i% }t^{2}}\,\mathrm{d}t$ , $x=0(.5)20(1)25$ , 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.

… ►

•

Zhang and Jin (1996, p. 642) includes the first 10 zeros of $\operatorname{erf}z$ , 9D; the first 25 distinct zeros of $C\left(z\right)$ and $S\left(z\right)$ , 8S.