exponential integrals

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21: 10.38 Derivatives with Respect to Order

… ►For the notations $E_1$ and $\mathrm{Ei}$ see §6.2(i). … ►

10.38.6 $${\frac{\partial I_{\nu }\left(x\right)}{\partial \nu }|}_{\nu =\pm \frac{1}{2}}=-\frac{1}{\sqrt{2\pi x}}\left(E_1\left(2x\right){\mathrm{e}}^x\pm \mathrm{Ei}\left(2x\right){\mathrm{e}}^{-x}\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{Ei}\left(x\right)$: exponential integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $x$: real variable and $\nu$: complex parameter

A&S Ref:

10.2.32 and 10.2.33 (both corrected)

Referenced by:

§10.38, Ch.10

Permalink:

http://dlmf.nist.gov/10.38.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.38, §10.38 and Ch.10

►

10.38.7 $${\frac{\partial K_{\nu }\left(x\right)}{\partial \nu }|}_{\nu =\pm \frac{1}{2}}=\pm \sqrt{\frac{\pi }{2x}}{E}_1\left(2x\right){\mathrm{e}}^x.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $x$: real variable and $\nu$: complex parameter

A&S Ref:

10.2.34 (corrected)

Referenced by:

§10.38, Ch.10

Permalink:

http://dlmf.nist.gov/10.38.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.38, §10.38 and Ch.10

…

22: 8.4 Special Values

… ►For $E_n\left(z\right)$ see §8.19(i). ►

8.4.1 $$\gamma (\frac{1}{2},z^2)=2{\int }_0^z{\mathrm{e}}^{-t^2}dt=\sqrt{\pi }\mathrm{erf}\left(z\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $\int$: integral and $z$: complex variable

A&S Ref:

6.5.16

Permalink:

http://dlmf.nist.gov/8.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §8.4 and Ch.8

… ►

8.4.4 $$\mathrm{\Gamma }(0,z)={\int }_z^{\mathrm{\infty }}{t}^{-1}{\mathrm{e}}^{-t}dt=E_1\left(z\right),$$

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\int$: integral and $z$: complex variable

A&S Ref:

6.5.15

Permalink:

http://dlmf.nist.gov/8.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §8.4 and Ch.8

… ►

8.4.13 $$\mathrm{\Gamma }(1-n,z)=z^{1-n}{E}_n\left(z\right),$$

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Symbols:

$E_p\left(z\right)$: generalized exponential integral, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§8.19(iv)

Permalink:

http://dlmf.nist.gov/8.4.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §8.4 and Ch.8

… ►

8.4.15 $$\mathrm{\Gamma }(-n,z)=\frac{{(-1)}^n}{n!}\left(E_1\left(z\right)-{\mathrm{e}}^{-z}\sum _{k=0}^{n-1}\frac{{(-1)}^{k}k!}{z^{k+1}}\right)=\frac{{(-1)}^n}{n!}\left(\psi \left(n+1\right)-\ln z\right)-z^{-n}\sum _{\begin{array}{c}k=0\\ k\ne n\end{array}}^{\mathrm{\infty }}\frac{{(-z)}^k}{k!(k-n)}.$$

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Symbols:

$\psi \left(z\right)$: psi (or digamma) function, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $!$: factorial (as in $n!$), $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\ln z$: principal branch of logarithm function, $z$: complex variable, $k$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

6.5.19

Referenced by:

§8.19(iii), §8.19(iv), §8.4

Permalink:

http://dlmf.nist.gov/8.4.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §8.4 and Ch.8

23: 6.17 Physical Applications

§6.17 Physical Applications

…

24: 8.28 Software

… ►

§8.28(vi) Generalized Exponential Integral for Real Argument and Integer Parameter

… ►

§8.28(vii) Generalized Exponential Integral for Complex Argument and/or Parameter

…

25: 6.12 Asymptotic Expansions

… ►

§6.12(i) Exponential and Logarithmic Integrals

►

6.12.1 $$E_1\left(z\right)\sim \frac{{\mathrm{e}}^{-z}}{z}\left(1-\frac{1!}{z}+\frac{2!}{z^2}-\frac{3!}{z^3}+\mathrm{\cdots }\right),$$ $z\to \mathrm{\infty }$, .

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase and $z$: complex variable

A&S Ref:

5.1.51 (special case of)

Permalink:

http://dlmf.nist.gov/6.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.12(i), §6.12 and Ch.6

… ►For these and other error bounds see Olver (1997b, pp. 109–112) with $\alpha =0$. … ►

6.12.2 $$\mathrm{Ei}\left(x\right)\sim \frac{{\mathrm{e}}^x}{x}\left(1+\frac{1!}{x}+\frac{2!}{x^2}+\frac{3!}{x^3}+\mathrm{\cdots }\right),$$ $x\to +\mathrm{\infty }$.

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $\mathrm{Ei}\left(x\right)$: exponential integral, $!$: factorial (as in $n!$) and $x$: real variable

Referenced by:

§6.12(i), §6.12(i)

Permalink:

http://dlmf.nist.gov/6.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.12(i), §6.12 and Ch.6

… ►

6.12.7 $$R_n^{(\mathrm{f})}(z)={(-1)}^n{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-zt}{t}^{2n}}{t^2+1}dt,$$

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Defines:

$R_n^{(\mathrm{f})}(z)$: remainder term (locally)

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§6.12(ii)

Permalink:

http://dlmf.nist.gov/6.12.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §6.12(ii), §6.12 and Ch.6

…

26: 6.13 Zeros

§6.13 Zeros

►The function $\mathrm{Ei}\left(x\right)$ has one real zero $x_0$, given by …

27: 2.11 Remainder Terms; Stokes Phenomenon

… ►From §8.19(i) the generalized exponential integral is given by … ►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 $E_p\left(z\right)$. For this reason the expansion of $E_p\left(z\right)$ in $|\mathrm{ph}z|\le \pi -\delta$ supplied by (2.11.8), (2.11.10), and (2.11.13) is said to be exponentially improved. … ►

2.11.25 $${\mathrm{e}}^5{E}_1\left(5\right)=0.20000-0.04000+0.01600-0.00960+0.00768-0.00768+0.00922-0.01290+0.02064-0.03716+0.07432-\mathrm{\cdots }.$$

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Symbols:

$\mathrm{e}$: base of natural logarithm and $E_1\left(z\right)$: exponential integral

Referenced by:

§2.11(vi)

Permalink:

http://dlmf.nist.gov/2.11.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §2.11(vi), §2.11 and Ch.2

… ►

2.11.26 $${\mathrm{e}}^5{E}_1\left(5\right)=0.17042\mathrm{\dots }.$$

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Symbols:

$\mathrm{e}$: base of natural logarithm and $E_1\left(z\right)$: exponential integral

Permalink:

http://dlmf.nist.gov/2.11.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §2.11(vi), §2.11 and Ch.2

…

28: 8.22 Mathematical Applications

§8.22 Mathematical Applications

… ►

8.22.1 $$F_p\left(z\right)=\frac{\mathrm{\Gamma }\left(p\right)}{2\pi }{z}^{1-p}{E}_p\left(z\right)=\frac{\mathrm{\Gamma }\left(p\right)}{2\pi }\mathrm{\Gamma }(1-p,z),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $E_p\left(z\right)$: generalized exponential integral, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $F_p\left(z\right)$: terminant function, $z$: complex variable and $p$: parameter

Permalink:

http://dlmf.nist.gov/8.22.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §8.22(i), §8.22 and Ch.8

… ►

8.22.2 $${\zeta }_x(s)=\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_0^x\frac{t^{s-1}}{{\mathrm{e}}^t-1}dt,$$ $\mathrm{\Re }s>1$,

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Defines:

${\zeta }_x(s)$: incomplete Riemann zeta function (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $x$: real variable

Permalink:

http://dlmf.nist.gov/8.22.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §8.22(ii), §8.22 and Ch.8

… ►The Debye functions ${\int }_0^x{t}^n{\left({\mathrm{e}}^t-1\right)}^{-1}dt$ and ${\int }_x^{\mathrm{\infty }}{t}^n{\left({\mathrm{e}}^t-1\right)}^{-1}dt$ are closely related to the incomplete Riemann zeta function and the Riemann zeta function. …

29: 8.1 Special Notation

… ►Unless otherwise indicated, primes denote derivatives with respect to the argument. ►The functions treated in this chapter are the incomplete gamma functions $\gamma (a,z)$, $\mathrm{\Gamma }(a,z)$, ${\gamma }^{\ast }(a,z)$, $P(a,z)$, and $Q(a,z)$; the incomplete beta functions ${\mathrm{B}}_x(a,b)$ and $I_x(a,b)$; the generalized exponential integral $E_p\left(z\right)$; the generalized sine and cosine integrals $\mathrm{si}(a,z)$, $\mathrm{ci}(a,z)$, $\mathrm{Si}(a,z)$, and $\mathrm{Ci}(a,z)$. …

30: 8.25 Methods of Computation

… ►Stable recursive schemes for the computation of $E_p\left(x\right)$ are described in Miller (1960) for $x>0$ and integer $p$. …