complementary exponential integral

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31: 7.13 Zeros

… ►

§7.13(ii) Zeros of $\mathrm{erfc}z$

… ►The other zeros of $\mathrm{erfc}z$ are ${\overline{z}}_n$. … ►In consequence of (7.5.5) and (7.5.10), zeros of $ℱ\left(z\right)$ are related to zeros of $\mathrm{erfc}z$. Thus if $z_n$ is a zero of $\mathrm{erfc}z$ (§7.13(ii)), then $(1+\mathrm{i})z_n/\sqrt{\pi }$ is a zero of $ℱ\left(z\right)$. ►For an asymptotic expansion of the zeros of ${\int }_0^z\exp \left(\frac{1}{2}\pi \mathrm{i}{t}^2\right)dt$ ($=ℱ\left(0\right)-ℱ\left(z\right)$ $=C\left(z\right)+\mathrm{i}S\left(z\right)$) see Tuẑilin (1971). …

32: 8.12 Uniform Asymptotic Expansions for Large Parameter

… ►

8.12.5 $$\frac{{\mathrm{e}}^{\pm \pi \mathrm{i}a}}{2\mathrm{i}\sin \left(\pi a\right)}Q(-a,z{\mathrm{e}}^{\pm \pi \mathrm{i}})=\pm \frac{1}{2}\mathrm{erfc}\left(\pm \mathrm{i}\eta \sqrt{a/2}\right)-\mathrm{i}T(a,\eta ),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $Q(a,z)$: normalized incomplete gamma function, $\sin z$: sine function, $z$: complex variable, $a$: parameter, $\eta (\lambda )$ and $T(a,\eta )$: function

Referenced by:

Erratum (V1.0.16) for Equation (8.12.5), Erratum (V1.0.16) for Equation (8.12.5)

Permalink:

http://dlmf.nist.gov/8.12.E5

Encodings:

pMML, png, TeX

Change (effective with 1.0.16):

To be consistent with the notation used in (8.12.16), the original equation,

$$\mathrm{\Gamma }\left(a+1\right)\frac{{\mathrm{e}}^{\pm \pi \mathrm{i}a}}{2\pi \mathrm{i}}\mathrm{\Gamma }(-a,z{\mathrm{e}}^{\pm \pi \mathrm{i}})=\mp \frac{1}{2}\mathrm{erfc}\left(\pm \mathrm{i}\eta \sqrt{a/2}\right)+\mathrm{i}T(a,\eta ),$$

was rewritten.

See also:

Annotations for §8.12 and Ch.8

… ►where $F\left(x\right)$ is Dawson’s integral; see §7.2(ii). … ►

$d(\pm \chi )=\sqrt{\frac{1}{2}\pi }{\mathrm{e}}^{{\chi }^2/2}\mathrm{erfc}\left(\pm \chi /\sqrt{2}\right),$

… ►For other uniform asymptotic approximations of the incomplete gamma functions in terms of the function $\mathrm{erfc}$ see Paris (2002b) and Dunster (1996a). … ►These expansions involve the inverse error function $\mathrm{inverfc}\left(x\right)$ (§7.17), and are uniform with respect to $q\in [0,1]$. …

33: 7.12 Asymptotic Expansions

… ►

§7.12(i) Complementary Error Function

… ►For these and other error bounds see Olver (1997b, pp. 109–112), with $\alpha =\frac{1}{2}$ and $z$ replaced by $z^2$; compare (7.11.2). … ►(Note that some of these re-expansions themselves involve the complementary error function.) ►

§7.12(ii) Fresnel Integrals

… ►

§7.12(iii) Goodwin–Staton Integral

…

34: 7.19 Voigt Functions

… ►

7.19.1 $$𝖴(x,t)=\frac{1}{\sqrt{4\pi t}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-{(x-y)}^2/(4t)}}{1+y^2}dy,$$

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Defines:

$𝖴(x,t)$: 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.E1

Encodings:

pMML, png, TeX

See also:

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

… ►

7.19.3 $$𝖴(x,t)+\mathrm{i}𝖵(x,t)=\sqrt{\frac{\pi }{4t}}{\mathrm{e}}^{z^2}\mathrm{erfc}z,$$ $z=(1-\mathrm{i}x)/(2\sqrt{t})$.

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Symbols:

$𝖴(x,t)$: Voigt function, $𝖵(x,t)$: Voigt function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable and $z$: complex variable

Referenced by:

§7.19(i), §7.22(iv)

Permalink:

http://dlmf.nist.gov/7.19.E3

Encodings:

pMML, png, TeX

See also:

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

… ►

§7.19(iv) Other Integral Representations

►

7.19.10 $$𝖴(\frac{u}{a},\frac{1}{4a^2})=a{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at-\frac{1}{4}{t}^2}\cos \left(ut\right)dt,$$

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Symbols:

$𝖴(x,t)$: Voigt function, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm and $\int$: integral

Referenced by:

§7.19(iii)

Permalink:

http://dlmf.nist.gov/7.19.E10

Encodings:

pMML, png, TeX

See also:

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

►

7.19.11 $$𝖵(\frac{u}{a},\frac{1}{4a^2})=a{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at-\frac{1}{4}{t}^2}\sin \left(ut\right)dt.$$

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Symbols:

$𝖵(x,t)$: Voigt function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\sin z$: sine function

Referenced by:

§7.19(iii)

Permalink:

http://dlmf.nist.gov/7.19.E11

Encodings:

pMML, png, TeX

See also:

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

35: 7.22 Methods of Computation

… ►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

… ►

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

►The recursion scheme given by (7.18.1) and (7.18.7) can be used for computing ${\mathrm{i}}^n\mathrm{erfc}\left(x\right)$. … ►The computation of these functions can be based on algorithms for the complementary error function with complex argument; compare (7.19.3). …

36: 8.11 Asymptotic Approximations and Expansions

… ►

8.11.10 $$P(a+1,x)=\frac{1}{2}\mathrm{erfc}\left(-y\right)-\frac{1}{3}\sqrt{\frac{2}{\pi a}}(1+y^2){\mathrm{e}}^{-y^2}+O\left(a^{-1}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $P(a,z)$: normalized incomplete gamma function, $x$: real variable, $a$: parameter and $y$: change of variable

Permalink:

http://dlmf.nist.gov/8.11.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §8.11(iv), §8.11 and Ch.8

►

8.11.11 $${\gamma }^{\ast }(1-a,-x)=x^{a-1}\left(-\cos \left(\pi a\right)+\frac{\sin \left(\pi a\right)}{\pi }\left(2\sqrt{\pi }F\left(y\right)+\frac{2}{3}\sqrt{\frac{2\pi }{a}}\left(1-y^2\right)\right){\mathrm{e}}^{y^2}+O\left(a^{-1}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $F\left(z\right)$: Dawson’s integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, ${\gamma }^{\ast }(a,z)$: incomplete gamma function, $\sin z$: sine function, $x$: real variable, $a$: parameter and $y$: change of variable

Permalink:

http://dlmf.nist.gov/8.11.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §8.11(iv), §8.11 and Ch.8

… ►For Dawson’s integral $F\left(y\right)$ see §7.2(ii). … ►This reference also contains explicit formulas for the coefficients in terms of Stirling numbers. … ►With $x=1$, an asymptotic expansion of $e_n(nx)/{\mathrm{e}}^{nx}$ follows from (8.11.14) and (8.11.16). …

37: 22.11 Fourier and Hyperbolic Series

… ►If , then … ►Next, if , then … ►Next, with $E=E\left(k\right)$ denoting the complete elliptic integral of the second kind (§19.2(ii)) and , … ►

22.11.14 $$k^2{\mathrm{sn}}^2(z,k)=\frac{E^{\prime }}{K^{\prime }}-{\left(\frac{\pi }{2K^{\prime }}\right)}^2\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\left({\mathrm{sech}}^2\left(\frac{\pi }{2K^{\prime }}(z-2nK)\right)\right),$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $E^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the second kind, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{sech}z$: hyperbolic secant function, $z$: complex and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.11.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §22.11 and Ch.22

►where $E^{\prime }=E^{\prime }\left(k\right)$ is defined by §19.2.9. …

38: 22.20 Methods of Computation

… ►If either $\tau$ or $q={\mathrm{e}}^{\mathrm{i}\pi \tau }$ is given, then we use $k={\theta }_2^2(0,q)/{\theta }_3^2(0,q)$, $k^{\prime }={\theta }_4^2(0,q)/{\theta }_3^2(0,q)$, $K=\frac{1}{2}\pi {\theta }_3^2(0,q)$, and $K^{\prime }=-\mathrm{i}\tau K$, obtaining the values of the theta functions as in §20.14. …

39: 2.11 Remainder Terms; Stokes Phenomenon

… ►From §8.19(i) the generalized exponential integral is given by …However, on combining (2.11.6) with the connection formula (8.19.18), with $m=1$, we derive … ►Owing to the factor ${\mathrm{e}}^{-\rho }$, that is, ${\mathrm{e}}^{-|z|}$ in (2.11.13), $F_{n+p}\left(z\right)$ is uniformly exponentially small compared with $E_p\left(z\right)$. … ►Here $\mathrm{erfc}$ is the complementary error function (§7.2(i)), and … ►A simple example is provided by Euler’s transformation (§3.9(ii)) applied to the asymptotic expansion for the exponential integral (§6.12(i)): …

40: 13.6 Relations to Other Functions

… ►

13.6.1 $$M(a,a,z)={\mathrm{e}}^z,$$

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable

A&S Ref:

13.6.12

Referenced by:

§13.10(v)

Permalink:

http://dlmf.nist.gov/13.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.6(i), §13.6 and Ch.13

►

13.6.2 $$M(1,2,2z)=\frac{{\mathrm{e}}^z}{z}\sinh z,$$

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function and $z$: complex variable

A&S Ref:

13.6.14

Permalink:

http://dlmf.nist.gov/13.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.6(i), §13.6 and Ch.13

… ►When $a-b$ is an integer or $a$ is a positive integer the Kummer functions can be expressed as incomplete gamma functions (or generalized exponential integrals). … ►

13.6.6 $$U(a,a,z)=z^{1-a}U(1,2-a,z)=z^{1-a}{\mathrm{e}}^z{E}_a\left(z\right)={\mathrm{e}}^z\mathrm{\Gamma }(1-a,z).$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\mathrm{\Gamma }(a,z)$: incomplete gamma function and $z$: complex variable

A&S Ref:

13.6.28 (in different form)

Permalink:

http://dlmf.nist.gov/13.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §13.6(ii), §13.6 and Ch.13

… ►

13.6.8 $$U(\frac{1}{2},\frac{1}{2},z^2)=\sqrt{\pi }{\mathrm{e}}^{z^2}\mathrm{erfc}\left(z\right).$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable

Referenced by:

§13.6(ii)

Permalink:

http://dlmf.nist.gov/13.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §13.6(ii), §13.6 and Ch.13

…