Goodwin%E2%80%93Staton%20integral

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2: 1.14 Integral Transforms

§1.14 Integral Transforms

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

f(t)

is absolutely integrable on

[0,R]

for every finite

R

, and the integral (1.14.47) converges, then … ►

§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: 7.22 Methods of Computation

… ►

§7.22(i) Main Functions

►The methods available for computing the main functions in this chapter are analogous to those described in §§6.18(i)–6.18(iv) for the exponential integral and sine and cosine integrals, and similar comments apply. … ►

§7.22(ii) Goodwin–Staton Integral

►See Goodwin and Staton (1948). ►

§7.22(iii) Repeated Integrals of the Complementary Error Function

…

4: 8.19 Generalized Exponential Integral

§8.19 Generalized Exponential Integral

►

§8.19(i) Definition and Integral Representations

… ►

Other Integral Representations

… ►

§8.19(ii) Graphics

… ►

§8.19(x) Integrals

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5: 6.2 Definitions and Interrelations

… ►

§6.2(i) Exponential and Logarithmic Integrals

… ► … ►The logarithmic integral is defined by … ►

§6.2(ii) Sine and Cosine Integrals

… ► …

6: 8.21 Generalized Sine and Cosine Integrals

§8.21 Generalized Sine and Cosine Integrals

… ►

§8.21(iii) Integral Representations

… ►

§8.21(iv) Interrelations

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§8.21(v) Special Values

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7: 7.25 Software

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§7.25(ii) $\operatorname{erf}x$ , $\operatorname{erfc}x$ , $\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(x\right)$ , $x\in\mathbb{R}$

… ►

§7.25(iv) $C\left(x\right)$ , $S\left(x\right)$ , $\mathrm{f}\left(x\right)$ , $\mathrm{g}\left(x\right)$ , $x\in\mathbb{R}$

… ►

§7.25(v) $C\left(z\right)$ , $S\left(z\right)$ , $z\in\mathbb{C}$

… ►

§7.25(vi) $\mathcal{F}\left(x\right)$ , $G\left(x\right)$ , $\mathsf{U}\left(x,t\right)$ , $\mathsf{V}\left(x,t\right)$ , $x\in\mathbb{R}$

… ►

§7.25(vii) $\mathcal{F}\left(z\right)$ , $G\left(z\right)$ , $z\in\mathbb{C}$

…

8: 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.

9: 7.5 Interrelations

§7.5 Interrelations

… ► … ►

7.5.13

G\left(x\right)=\sqrt{\pi}F\left(x\right)-\tfrac{1}{2}e^{-x^{2}}\operatorname{% Ei}\left(x^{2}\right),

x>0

ⓘ

Symbols:: $F\left(\NVar{z}\right)$ : Dawson’s integral, $G\left(\NVar{z}\right)$ : Goodwin–Staton integral, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\operatorname{Ei}\left(\NVar{x}\right)$ : exponential integral and $x$ : real variable
Referenced by:: §7.5
Permalink:: http://dlmf.nist.gov/7.5.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §7.5 and Ch.7

►For

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

see §6.2(i).

10: 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