logarithmic integral

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21: 6 Exponential, Logarithmic, Sine, and
Cosine Integrals

Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals

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22: 8.20 Asymptotic Expansions of $E_p\left(z\right)$

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8.20.1 $$E_p\left(z\right)=\frac{{\mathrm{e}}^{-z}}{z}\left(\sum _{k=0}^{n-1}{(-1)}^k\frac{{\left(p\right)}_k}{z^k}+{(-1)}^n\frac{{\left(p\right)}_n{\mathrm{e}}^z}{z^{n-1}}{E}_{n+p}\left(z\right)\right),$$ $n=1,2,3,\mathrm{\dots }$.

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $z$: complex variable, $p$: parameter, $k$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

5.1.51

Permalink:

http://dlmf.nist.gov/8.20.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §8.20(i), §8.20 and Ch.8

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8.20.2 $$E_p\left(z\right)\sim \frac{{\mathrm{e}}^{-z}}{z}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{{\left(p\right)}_k}{z^k},$$ $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$,

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\mathrm{ph}$: phase, $z$: complex variable, $p$: parameter, $k$: nonnegative integer and $\delta$: small positive constant

Referenced by:

§8.20(i)

Permalink:

http://dlmf.nist.gov/8.20.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §8.20(i), §8.20 and Ch.8

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8.20.3 $$E_p\left(z\right)\sim \pm \frac{2\pi \mathrm{i}}{\mathrm{\Gamma }\left(p\right)}{\mathrm{e}}^{\mp p\pi \mathrm{i}}{z}^{p-1}+\frac{{\mathrm{e}}^{-z}}{z}\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{\left(p\right)}_k}{z^k},$$ $\frac{1}{2}\pi +\delta \le \pm \mathrm{ph}z\le \frac{7}{2}\pi -\delta$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, $z$: complex variable, $p$: parameter, $k$: nonnegative integer and $\delta$: small positive constant

Referenced by:

§2.11(ii), §8.20(i)

Permalink:

http://dlmf.nist.gov/8.20.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §8.20(i), §8.20 and Ch.8

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8.20.6 $$E_p\left(\lambda p\right)\sim \frac{{\mathrm{e}}^{-\lambda p}}{(\lambda +1)p}\sum _{k=0}^{\mathrm{\infty }}\frac{A_k(\lambda )}{{(\lambda +1)}^{2k}}\frac{1}{p^k},$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $p$: parameter, $k$: nonnegative integer and $A_k(\lambda )$

Permalink:

http://dlmf.nist.gov/8.20.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §8.20(ii), §8.20 and Ch.8

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23: 6.10 Other Series Expansions

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6.10.1 $$E_1\left(z\right)={\mathrm{e}}^{-z}\left(\frac{c_0}{z}+\frac{c_1}{z(z+1)}+\frac{2!c_2}{z(z+1)(z+2)}+\frac{3!c_3}{z(z+1)(z+2)(z+3)}+\mathrm{\cdots }\right),$$ $\mathrm{\Re }z>0$,

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Symbols:

$\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $!$: factorial (as in $n!$), $\mathrm{\Re }$: real part and $z$: complex variable

Permalink:

http://dlmf.nist.gov/6.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.10(i), §6.10 and Ch.6

… ►For a more general result (incomplete gamma function), and also for a result for the logarithmic integral, see Nielsen (1906a, p. 283: Formula (3) is incorrect). … ►

6.10.6 $$\mathrm{Ei}\left(x\right)=\gamma +\ln \left|x\right|+\sum _{n=0}^{\mathrm{\infty }}{(-1)}^n(x-a_n){\left({𝗂}_n^{(1)}\left(\frac{1}{2}x\right)\right)}^2,$$ $x\ne 0$,

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Symbols:

$\gamma$: Euler’s constant, $\mathrm{Ei}\left(x\right)$: exponential integral, $\ln z$: principal branch of logarithm function, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $x$: real variable, $n$: nonnegative integer and $a_n$: coefficients

Referenced by:

§6.18(i)

Permalink:

http://dlmf.nist.gov/6.10.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §6.10(ii), §6.10 and Ch.6

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6.10.8 $$\mathrm{Ein}\left(z\right)=z{\mathrm{e}}^{-z/2}\left({𝗂}_0^{(1)}\left(\frac{1}{2}z\right)+\sum _{n=1}^{\mathrm{\infty }}\frac{2n+1}{n(n+1)}{𝗂}_n^{(1)}\left(\frac{1}{2}z\right)\right).$$

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Symbols:

$\mathrm{Ein}\left(z\right)$: complementary exponential integral, $\mathrm{e}$: base of natural logarithm, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§6.10(ii)

Permalink:

http://dlmf.nist.gov/6.10.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §6.10(ii), §6.10 and Ch.6

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24: 12.12 Integrals

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12.12.1 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{1}{4}{t}^2}{t}^{\mu -1}U(a,t)dt=\frac{\sqrt{\pi }{2}^{-\frac{1}{2}(\mu +a+\frac{1}{2})}\mathrm{\Gamma }\left(\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}(\mu +a+\frac{3}{2})\right)},$$ $\mathrm{\Re }\mu >0$ ,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $U(a,z)$: parabolic cylinder function, $\mathrm{\Re }$: real part and $a$: real or complex parameter

Permalink:

http://dlmf.nist.gov/12.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §12.12 and Ch.12

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12.12.2 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{3}{4}{t}^2}{t}^{-a-\frac{3}{2}}U(a,t)dt=2^{\frac{1}{4}+\frac{1}{2}a}\mathrm{\Gamma }\left(-a-\frac{1}{2}\right)\cos \left((\frac{1}{4}a+\frac{1}{8})\pi \right),$$ ,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $U(a,z)$: parabolic cylinder function, $\mathrm{\Re }$: real part and $a$: real or complex parameter

Permalink:

http://dlmf.nist.gov/12.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §12.12 and Ch.12

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12.12.3 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{1}{4}{t}^2}{t}^{-a-\frac{1}{2}}{(x^2+t^2)}^{-1}U(a,t)dt=\sqrt{\pi /2}\mathrm{\Gamma }\left(\frac{1}{2}-a\right)x^{-a-\frac{3}{2}}{\mathrm{e}}^{\frac{1}{4}{x}^2}U(-a,x),$$ .

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $U(a,z)$: parabolic cylinder function, $\mathrm{\Re }$: real part, $x$: real variable and $a$: real or complex parameter

Permalink:

http://dlmf.nist.gov/12.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §12.12 and Ch.12

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12.12.4 $${(U(a,z))}^2+{(\overline{U}(a,z))}^2=\frac{2^{\frac{3}{2}}}{\pi }\mathrm{\Gamma }\left(\frac{1}{2}-a\right){\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{2at+\frac{1}{2}{z}^2\tanh t}}{\sqrt{\sinh \left(2t\right)}}dt,$$ .

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\tanh z$: hyperbolic tangent function, $\int$: integral, $U(a,z)$: parabolic cylinder function, $\overline{U}(a,x)$: parabolic cylinder function, $\mathrm{\Re }$: real part, $z$: complex variable and $a$: real or complex parameter

Notes:

See Durand (1975).

Referenced by:

§12.12

Permalink:

http://dlmf.nist.gov/12.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.12, §12.12 and Ch.12

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25: 11.7 Integrals and Sums

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11.7.13 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}{𝐇}_0\left(t\right)dt=\frac{2}{\pi \sqrt{1+a^2}}\ln \left(\frac{1+\sqrt{1+a^2}}{a}\right),$$

ⓘ

Symbols:

${𝐇}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\ln z$: principal branch of logarithm function and $a$: parameter

Permalink:

http://dlmf.nist.gov/11.7.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §11.7(iii), §11.7 and Ch.11

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11.7.14 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}{𝐇}_1\left(t\right)dt=\frac{2}{\pi a}-\frac{2a}{\pi \sqrt{1+a^2}}\ln \left(\frac{1+\sqrt{1+a^2}}{a}\right),$$

ⓘ

Symbols:

${𝐇}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\ln z$: principal branch of logarithm function and $a$: parameter

Permalink:

http://dlmf.nist.gov/11.7.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §11.7(iii), §11.7 and Ch.11

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11.7.15 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}{𝐋}_0\left(t\right)dt=\frac{2}{\pi \sqrt{a^2-1}}\arcsin \left(\frac{1}{a}\right),$$

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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, $\arcsin z$: arcsine function, ${𝐋}_{\nu }\left(z\right)$: modified Struve function and $a$: parameter

Permalink:

http://dlmf.nist.gov/11.7.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §11.7(iii), §11.7 and Ch.11

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26: 7.2 Definitions

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7.2.1 $$\mathrm{erf}z=\frac{2}{\sqrt{\pi }}{\int }_0^z{\mathrm{e}}^{-t^2}dt,$$

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Defines:

$\mathrm{erf}z$: error 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 $z$: complex variable

A&S Ref:

7.1.1

Referenced by:

§7.5, §7.6(ii)

Permalink:

http://dlmf.nist.gov/7.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §7.2(i), §7.2 and Ch.7

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7.2.2 $$\mathrm{erfc}z=\frac{2}{\sqrt{\pi }}{\int }_z^{\mathrm{\infty }}{\mathrm{e}}^{-t^2}dt=1-\mathrm{erf}z,$$

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Defines:

$\mathrm{erfc}z$: complementary error function

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, $\int$: integral and $z$: complex variable

A&S Ref:

7.1.2

Referenced by:

§7.5, §7.7(i)

Permalink:

http://dlmf.nist.gov/7.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §7.2(i), §7.2 and Ch.7

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7.2.5 $$F\left(z\right)={\mathrm{e}}^{-z^2}{\int }_0^z{\mathrm{e}}^{t^2}dt.$$

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Defines:

$F\left(z\right)$: Dawson’s integral

Symbols:

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

A&S Ref:

7.1.1

Permalink:

http://dlmf.nist.gov/7.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §7.2(ii), §7.2 and Ch.7

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7.2.6 $$ℱ\left(z\right)={\int }_z^{\mathrm{\infty }}{\mathrm{e}}^{\frac{1}{2}\pi \mathrm{i}{t}^2}dt,$$

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Defines:

$ℱ\left(z\right)$: Fresnel integral

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral and $z$: complex variable

Referenced by:

§7.5, §7.7(ii)

Permalink:

http://dlmf.nist.gov/7.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §7.2(iii), §7.2 and Ch.7

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7.2.12 $$G\left(z\right)={\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-t^2}}{t+z}dt,$$ .

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Defines:

$G\left(z\right)$: Goodwin–Staton integral

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, $\mathrm{ph}$: phase and $z$: complex variable

A&S Ref:

27.6 (in different notation)

Permalink:

http://dlmf.nist.gov/7.2.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §7.2(v), §7.2 and Ch.7

27: 13.16 Integral Representations

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13.16.1 $$M_{\kappa ,\mu }\left(z\right)=\frac{\mathrm{\Gamma }\left(1+2\mu \right)z^{\mu +\frac{1}{2}}{2}^{-2\mu }}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)\mathrm{\Gamma }\left(\frac{1}{2}+\mu +\kappa \right)}{\int }_{-1}^1{\mathrm{e}}^{\frac{1}{2}zt}{(1+t)}^{\mu -\frac{1}{2}-\kappa }{(1-t)}^{\mu -\frac{1}{2}+\kappa }dt,$$ $\mathrm{\Re }\mu +\frac{1}{2}>\left|\mathrm{\Re }\kappa \right|$,

ⓘ

Symbols:

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

Referenced by:

§13.29(iii)

Permalink:

http://dlmf.nist.gov/13.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.16(i), §13.16 and Ch.13

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13.16.2 $$M_{\kappa ,\mu }\left(z\right)=\frac{\mathrm{\Gamma }\left(1+2\mu \right)z^{\lambda }}{\mathrm{\Gamma }\left(1+2\mu -2\lambda \right)\mathrm{\Gamma }\left(2\lambda \right)}{\int }_0^1{M}_{\kappa -\lambda ,\mu -\lambda }\left(zt\right){\mathrm{e}}^{\frac{1}{2}z(t-1)}{t}^{\mu -\lambda -\frac{1}{2}}{(1-t)}^{2\lambda -1}dt,$$ $\mathrm{\Re }\mu +\frac{1}{2}>\mathrm{\Re }\lambda >0$,

ⓘ

Symbols:

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

Referenced by:

§13.23(v)

Permalink:

http://dlmf.nist.gov/13.16.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.16(i), §13.16 and Ch.13

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13.16.3 $$\frac{1}{\mathrm{\Gamma }\left(1+2\mu \right)}{M}_{\kappa ,\mu }\left(z\right)=\frac{\sqrt{z}{\mathrm{e}}^{\frac{1}{2}z}}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu +\kappa \right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{\kappa -\frac{1}{2}}{J}_{2\mu }\left(2\sqrt{zt}\right)dt,$$ $\mathrm{\Re }\left(\kappa +\mu \right)+\frac{1}{2}>0$,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.16.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.16(i), §13.16 and Ch.13

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13.16.4 $$\frac{1}{\mathrm{\Gamma }\left(1+2\mu \right)}{M}_{\kappa ,\mu }\left(z\right)=\frac{\sqrt{z}{\mathrm{e}}^{-\frac{1}{2}z}}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{-\kappa -\frac{1}{2}}{I}_{2\mu }\left(2\sqrt{zt}\right)dt,$$ .

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

Erratum (V1.0.5) for Equation (13.16.4)

Permalink:

http://dlmf.nist.gov/13.16.E4

Encodings:

pMML, png, TeX

Errata (effective with 1.0.5):

Originally the condition for the validity of this equation was stated incorrectly as $\mathrm{\Re }(\kappa -\mu )-\frac{1}{2}>0$. The correct condition is .

See also:

Annotations for §13.16(i), §13.16 and Ch.13

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13.16.5 $$W_{\kappa ,\mu }\left(z\right)=\frac{z^{\mu +\frac{1}{2}}{2}^{-2\mu }}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)}{\int }_1^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{1}{2}zt}{(t-1)}^{\mu -\frac{1}{2}-\kappa }{(t+1)}^{\mu -\frac{1}{2}+\kappa }dt,$$ $\mathrm{\Re }\mu +\frac{1}{2}>\mathrm{\Re }\kappa$, ,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

§13.29(iii)

Permalink:

http://dlmf.nist.gov/13.16.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §13.16(i), §13.16 and Ch.13

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28: 6.11 Relations to Other Functions

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6.11.2 $$E_1\left(z\right)={\mathrm{e}}^{-z}U(1,1,z),$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral and $z$: complex variable

Referenced by:

§6.11, 5th item, §6.6

Permalink:

http://dlmf.nist.gov/6.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.11, §6.11 and Ch.6

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29: 4.26 Integrals

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4.26.3 $$\int \tan xdx=-\ln \left(\cos x\right),$$ .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, $\tan z$: tangent function and $x$: real variable

A&S Ref:

4.3.115

Permalink:

http://dlmf.nist.gov/4.26.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.26(ii), §4.26 and Ch.4

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4.26.4 $$\int \csc xdx=\ln \left(\tan \frac{1}{2}x\right),$$ .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, $\tan z$: tangent function and $x$: real variable

A&S Ref:

4.3.116

Permalink:

http://dlmf.nist.gov/4.26.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.26(ii), §4.26 and Ch.4

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4.26.6 $$\int \cot xdx=\ln \left(\sin x\right),$$ .

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, $\sin z$: sine function and $x$: real variable

A&S Ref:

4.3.118

Permalink:

http://dlmf.nist.gov/4.26.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.26(ii), §4.26 and Ch.4

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4.26.7 $$\int {\mathrm{e}}^{ax}\sin \left(bx\right)dx=\frac{{\mathrm{e}}^{ax}}{a^2+b^2}(a\sin \left(bx\right)-b\cos \left(bx\right)),$$

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Symbols:

$\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\sin z$: sine function, $a$: real or complex constant and $x$: real variable

A&S Ref:

4.3.136

Permalink:

http://dlmf.nist.gov/4.26.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §4.26(ii), §4.26 and Ch.4

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4.26.8 $$\int {\mathrm{e}}^{ax}\cos \left(bx\right)dx=\frac{{\mathrm{e}}^{ax}}{a^2+b^2}(a\cos \left(bx\right)+b\sin \left(bx\right)).$$

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Symbols:

$\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\sin z$: sine function, $a$: real or complex constant and $x$: real variable

A&S Ref:

4.3.137

Permalink:

http://dlmf.nist.gov/4.26.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §4.26(ii), §4.26 and Ch.4

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30: 7.8 Inequalities

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7.8.1 $$𝖬\left(x\right)=\frac{{\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{-t^2}dt}{{\mathrm{e}}^{-x^2}}={\mathrm{e}}^{x^2}{\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{-t^2}dt.$$

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Defines:

$𝖬\left(x\right)$: Mills’ ratio

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $x$: real variable

Permalink:

http://dlmf.nist.gov/7.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §7.8 and Ch.7

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7.8.6 $a,x>0$.

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $x$: real variable

Referenced by:

§7.8

Permalink:

http://dlmf.nist.gov/7.8.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §7.8 and Ch.7

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7.8.7 $x>0$.

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Symbols:

$F\left(z\right)$: Dawson’s integral, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\int$: integral and $x$: real variable

Referenced by:

Erratum (V1.1.5) for Additions

Permalink:

http://dlmf.nist.gov/7.8.E7

Encodings:

pMML, png, TeX

Addition (effective with 1.1.5):

A new inequality was added.

See also:

Annotations for §7.8 and Ch.7

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