logarithmic integral

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11: 22.14 Integrals

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22.14.7 $$\int \mathrm{dc}(x,k)dx=\ln \left(\mathrm{nc}(x,k)+\mathrm{sc}(x,k)\right),$$

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Symbols:

$\mathrm{dc}(z,k)$: Jacobian elliptic function, $\mathrm{nc}(z,k)$: Jacobian elliptic function, $\mathrm{sc}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, $x$: real and $k$: modulus

A&S Ref:

16.24.7

Permalink:

http://dlmf.nist.gov/22.14.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §22.14(i), §22.14 and Ch.22

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22.14.10 $$\int \mathrm{ns}(x,k)dx=\ln \left(\mathrm{ds}(x,k)-\mathrm{cs}(x,k)\right),$$

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Symbols:

$\mathrm{cs}(z,k)$: Jacobian elliptic function, $\mathrm{ds}(z,k)$: Jacobian elliptic function, $\mathrm{ns}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, $x$: real and $k$: modulus

A&S Ref:

16.24.10

Permalink:

http://dlmf.nist.gov/22.14.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §22.14(i), §22.14 and Ch.22

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22.14.11 $$\int \mathrm{ds}(x,k)dx=\ln \left(\mathrm{ns}(x,k)-\mathrm{cs}(x,k)\right),$$

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Symbols:

$\mathrm{cs}(z,k)$: Jacobian elliptic function, $\mathrm{ds}(z,k)$: Jacobian elliptic function, $\mathrm{ns}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, $x$: real and $k$: modulus

A&S Ref:

16.24.11

Permalink:

http://dlmf.nist.gov/22.14.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §22.14(i), §22.14 and Ch.22

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22.14.12 $$\int \mathrm{cs}(x,k)dx=\ln \left(\mathrm{ns}(x,k)-\mathrm{ds}(x,k)\right).$$

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Symbols:

$\mathrm{cs}(z,k)$: Jacobian elliptic function, $\mathrm{ds}(z,k)$: Jacobian elliptic function, $\mathrm{ns}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, $x$: real and $k$: modulus

A&S Ref:

16.24.12

Permalink:

http://dlmf.nist.gov/22.14.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §22.14(i), §22.14 and Ch.22

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22.14.18 $${\int }_0^{K\left(k\right)}\ln \left(\mathrm{dn}(t,k)\right)dt=\frac{1}{2}K\left(k\right)\ln k^{\prime }.$$

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Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, $k$: modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/22.14.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §22.14(iv), §22.14 and Ch.22

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12: 6.15 Sums

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6.15.1 $$\sum _{n=1}^{\mathrm{\infty }}\mathrm{Ci}\left(\pi n\right)=\frac{1}{2}(\ln 2-\gamma ),$$

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Symbols:

$\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ci}\left(z\right)$: cosine integral, $\ln z$: principal branch of logarithm function and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/6.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.15 and Ch.6

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6.15.2 $$\sum _{n=1}^{\mathrm{\infty }}\frac{\mathrm{si}\left(\pi n\right)}{n}=\frac{1}{2}\pi (\ln \pi -1),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $\mathrm{si}\left(z\right)$: sine integral and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/6.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.15 and Ch.6

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6.15.3 $$\sum _{n=1}^{\mathrm{\infty }}{(-1)}^n\mathrm{Ci}\left(2\pi n\right)=1-\ln 2-\frac{1}{2}\gamma ,$$

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Symbols:

$\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ci}\left(z\right)$: cosine integral, $\ln z$: principal branch of logarithm function and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/6.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.15 and Ch.6

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6.15.4 $$\sum _{n=1}^{\mathrm{\infty }}{(-1)}^n\frac{\mathrm{si}\left(2\pi n\right)}{n}=\pi (\frac{3}{2}\ln 2-1).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $\mathrm{si}\left(z\right)$: sine integral and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/6.15.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §6.15 and Ch.6

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13: 6.9 Continued Fraction

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6.9.1 $$E_1\left(z\right)=\frac{{\mathrm{e}}^{-z}}{z+}\frac{1}{1+}\frac{1}{z+}\frac{2}{1+}\frac{2}{z+}\frac{3}{1+}\frac{3}{z+}\mathrm{\cdots },$$ .

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

A&S Ref:

5.1.22 (special case of)

Referenced by:

§6.18(i)

Permalink:

http://dlmf.nist.gov/6.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.9 and Ch.6

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14: 6.4 Analytic Continuation

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6.4.1 $$E_1\left(z\right)=\mathrm{Ein}\left(z\right)-\mathrm{Ln}z-\gamma ;$$

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Symbols:

$\gamma$: Euler’s constant, $\mathrm{Ein}\left(z\right)$: complementary exponential integral, $E_1\left(z\right)$: exponential integral, $\mathrm{Ln}z$: general logarithm function and $z$: complex variable

Referenced by:

§6.4

Permalink:

http://dlmf.nist.gov/6.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.4 and Ch.6

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6.4.2 $$E_1\left(z{\mathrm{e}}^{2m\pi \mathrm{i}}\right)=E_1\left(z\right)-2m\pi \mathrm{i},$$ $m\in \mathbb{Z}$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{i}$: imaginary unit, $\mathbb{Z}$: set of all integers and $z$: complex variable

Permalink:

http://dlmf.nist.gov/6.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.4 and Ch.6

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6.4.3 $$E_1\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)=\mathrm{Ein}\left(-z\right)-\ln z-\gamma \mp \pi \mathrm{i},$$ $|\mathrm{ph}z|\le \pi$.

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Symbols:

$\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ein}\left(z\right)$: complementary exponential integral, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase and $z$: complex variable

Referenced by:

§6.4

Permalink:

http://dlmf.nist.gov/6.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.4 and Ch.6

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6.4.5 $$\mathrm{Chi}\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)=\pm \pi \mathrm{i}+\mathrm{Chi}\left(z\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{Chi}\left(z\right)$: hyperbolic cosine integral, $\mathrm{i}$: imaginary unit and $z$: complex variable

Referenced by:

§6.4

Permalink:

http://dlmf.nist.gov/6.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §6.4 and Ch.6

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6.4.6 $$\mathrm{f}\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)=\pi {\mathrm{e}}^{\mp \mathrm{i}z}-\mathrm{f}\left(z\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

Referenced by:

§6.12(ii), §6.4

Permalink:

http://dlmf.nist.gov/6.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §6.4 and Ch.6

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15: 5.9 Integral Representations

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Binet’s Formula

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5.9.10_2 $$\mathrm{Ln}\mathrm{\Gamma }\left(z\right)=\left(z-\frac{1}{2}\right)\ln z-z+\frac{1}{2}\ln \left(2\pi \right)+{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}\left(\frac{1}{{\mathrm{e}}^t-1}-\frac{1}{t}+\frac{1}{2}\right)\frac{dt}{t},$$

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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, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function and $z$: complex variable

Source:

Whittaker and Watson (1927, §12.31)

Referenced by:

§5.9(i), Erratum (V1.1.7) for Additions

Permalink:

http://dlmf.nist.gov/5.9.E10_2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

See also:

Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

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5.9.13 $$\psi \left(z\right)=\ln z+{\int }_0^{\mathrm{\infty }}\left(\frac{1}{t}-\frac{1}{1-{\mathrm{e}}^{-t}}\right){\mathrm{e}}^{-tz}dt,$$

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Symbols:

$dx$: differential of $x$ , $\psi \left(z\right)$: psi (or digamma) function, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\ln z$: principal branch of logarithm function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/5.9.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §5.9(ii), §5.9 and Ch.5

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5.9.15 $$\psi \left(z\right)=\ln z-\frac{1}{2z}-2{\int }_0^{\mathrm{\infty }}\frac{tdt}{(t^2+z^2)({\mathrm{e}}^{2\pi t}-1)}.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\psi \left(z\right)$: psi (or digamma) function, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

6.3.21

Referenced by:

§5.9(ii)

Permalink:

http://dlmf.nist.gov/5.9.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §5.9(ii), §5.9 and Ch.5

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5.9.19 $${\mathrm{\Gamma }}^{(n)}\left(z\right)={\int }_0^{\mathrm{\infty }}{(\ln t)}^n{\mathrm{e}}^{-t}{t}^{z-1}dt,$$ $n\ge 0$, $\mathrm{\Re }z>0$.

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Symbols:

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

Referenced by:

§5.9(ii)

Permalink:

http://dlmf.nist.gov/5.9.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §5.9(ii), §5.9 and Ch.5

16: 4.40 Integrals

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4.40.3 $$\int \tanh xdx=\ln \left(\cosh x\right).$$

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Symbols:

$dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $\int$: integral, $\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

4.5.79

Permalink:

http://dlmf.nist.gov/4.40.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(ii), §4.40 and Ch.4

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4.40.4 $$\int \mathrm{csch}xdx=\ln \left(\tanh \left(\frac{1}{2}x\right)\right),$$ .

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Symbols:

$dx$: differential of $x$ , $\mathrm{csch}z$: hyperbolic cosecant function, $\tanh z$: hyperbolic tangent function, $\int$: integral, $\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

4.5.80

Permalink:

http://dlmf.nist.gov/4.40.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(ii), §4.40 and Ch.4

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4.40.6 $$\int \coth xdx=\ln \left(\sinh x\right),$$ .

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Symbols:

$dx$: differential of $x$ , $\coth z$: hyperbolic cotangent function, $\sinh z$: hyperbolic sine function, $\int$: integral, $\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

4.5.82

Permalink:

http://dlmf.nist.gov/4.40.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(ii), §4.40 and Ch.4

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4.40.13 $$\int \mathrm{arctanh}xdx=x\mathrm{arctanh}x+\frac{1}{2}\ln \left(1-x^2\right),$$ ,

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Symbols:

$dx$: differential of $x$ , $\mathrm{arctanh}z$: inverse hyperbolic tangent function, $\int$: integral, $\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

4.6.45 (modified)

Permalink:

http://dlmf.nist.gov/4.40.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(iv), §4.40 and Ch.4

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4.40.16 $$\int \mathrm{arccoth}xdx=x\mathrm{arccoth}x+\frac{1}{2}\ln \left(x^2-1\right),$$ .

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Symbols:

$dx$: differential of $x$ , $\mathrm{arccoth}z$: inverse hyperbolic cotangent function, $\int$: integral, $\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

4.6.48 (modified)

Permalink:

http://dlmf.nist.gov/4.40.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(iv), §4.40 and Ch.4

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17: 8.19 Generalized Exponential Integral

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8.19.5 $$E_0\left(z\right)=z^{-1}{\mathrm{e}}^{-z},$$ $z\ne 0$,

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Symbols:

$\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral and $z$: complex variable

A&S Ref:

5.1.24

Permalink:

http://dlmf.nist.gov/8.19.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(iii), §8.19 and Ch.8

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8.19.12 $$p{E}_{p+1}\left(z\right)+z{E}_p\left(z\right)={\mathrm{e}}^{-z}.$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $z$: complex variable and $p$: parameter

Permalink:

http://dlmf.nist.gov/8.19.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(v), §8.19 and Ch.8

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8.19.15 $$\frac{{\partial }^j{E}_p\left(z\right)}{{\partial p}^j}={(-1)}^j{\int }_1^{\mathrm{\infty }}{(\ln t)}^j{t}^{-p}{\mathrm{e}}^{-zt}dt,$$ $\mathrm{\Re }z>0$.

ⓘ

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\int$: integral, $\ln z$: principal branch of logarithm function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\mathrm{\Re }$: real part, $z$: complex variable, $p$: parameter and $j$: numbers

Permalink:

http://dlmf.nist.gov/8.19.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(v), §8.19(v), §8.19 and Ch.8

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8.19.16 $$E_p\left(z\right)=z^{p-1}{\mathrm{e}}^{-z}U(p,p,z).$$

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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, $z$: complex variable and $p$: parameter

Permalink:

http://dlmf.nist.gov/8.19.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(vi), §8.19 and Ch.8

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8.19.24 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}{E}_n\left(t\right)dt=\frac{{(-1)}^{n-1}}{a^n}\left(\ln \left(1+a\right)+\sum _{k=1}^{n-1}\frac{{(-1)}^k{a}^k}{k}\right),$$ $n=1,2,\mathrm{\dots }$, $\mathrm{\Re }a>-1$,

ⓘ

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\int$: integral, $\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part, $a$: parameter, $k$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

5.1.34

Referenced by:

§8.19(x)

Permalink:

http://dlmf.nist.gov/8.19.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(x), §8.19 and Ch.8

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18: 7.7 Integral Representations

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7.7.9 $${\int }_0^x\mathrm{erf}tdt=x\mathrm{erf}x+\frac{1}{\sqrt{\pi }}\left({\mathrm{e}}^{-x^2}-1\right).$$

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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, $\int$: integral and $x$: real variable

A&S Ref:

7.4.35 (in different form)

Referenced by:

§7.7(i)

Permalink:

http://dlmf.nist.gov/7.7.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(i), §7.7 and Ch.7

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7.7.10 $$\mathrm{f}\left(z\right)=\frac{1}{\pi \sqrt{2}}{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-\pi z^{2}t/2}}{\sqrt{t}(t^2+1)}dt,$$ $|\mathrm{ph}z|\le \frac{1}{4}\pi$,

ⓘ

Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\pi$: 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:

7.4.26 (in different form)

Referenced by:

§7.12(ii), §7.7(ii)

Permalink:

http://dlmf.nist.gov/7.7.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(ii), §7.7 and Ch.7

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7.7.11 $$\mathrm{g}\left(z\right)=\frac{1}{\pi \sqrt{2}}{\int }_0^{\mathrm{\infty }}\frac{\sqrt{t}{\mathrm{e}}^{-\pi z^{2}t/2}}{t^2+1}dt,$$ $|\mathrm{ph}z|\le \frac{1}{4}\pi$,

ⓘ

Symbols:

$\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $\pi$: 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:

7.4.25 (in different form)

Referenced by:

§7.12(ii), §7.7(ii)

Permalink:

http://dlmf.nist.gov/7.7.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(ii), §7.7 and Ch.7

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7.7.15 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}\cos \left(t^2\right)dt=\sqrt{\frac{\pi }{2}}\mathrm{f}\left(\frac{a}{\sqrt{2\pi }}\right),$$ $\mathrm{\Re }a>0$,

ⓘ

Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\pi$: 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 and $\mathrm{\Re }$: real part

Keywords:

Laplace transform

A&S Ref:

7.4.22 (in different notation)

Referenced by:

§7.14(ii)

Permalink:

http://dlmf.nist.gov/7.7.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(ii), §7.7(ii), §7.7 and Ch.7

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7.7.16 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}\sin \left(t^2\right)dt=\sqrt{\frac{\pi }{2}}\mathrm{g}\left(\frac{a}{\sqrt{2\pi }}\right),$$ $\mathrm{\Re }a>0$.

ⓘ

Symbols:

$\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $\sin z$: sine function

Keywords:

Laplace transform

A&S Ref:

7.4.23 (in different notation)

Referenced by:

§7.14(ii)

Permalink:

http://dlmf.nist.gov/7.7.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(ii), §7.7(ii), §7.7 and Ch.7

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19: 7.14 Integrals

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7.14.3 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}\mathrm{erf}\sqrt{bt}dt=\frac{1}{a}\sqrt{\frac{b}{a+b}},$$ $\mathrm{\Re }a>0$, $\mathrm{\Re }b>0$,

ⓘ

Symbols:

$dx$: differential of $x$ , $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Laplace transform

A&S Ref:

7.4.19

Referenced by:

§7.14(i)

Permalink:

http://dlmf.nist.gov/7.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §7.14(i), §7.14(i), §7.14 and Ch.7

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7.14.5 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}C\left(t\right)dt=\frac{1}{a}\mathrm{f}\left(\frac{a}{\pi }\right),$$ $\mathrm{\Re }a>0$,

ⓘ

Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $C\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Laplace transform

A&S Ref:

7.4.27 (in different notation)

Referenced by:

§7.14(ii)

Permalink:

http://dlmf.nist.gov/7.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §7.14(ii), §7.14(ii), §7.14 and Ch.7

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7.14.6 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}S\left(t\right)dt=\frac{1}{a}\mathrm{g}\left(\frac{a}{\pi }\right),$$ $\mathrm{\Re }a>0$,

ⓘ

Symbols:

$\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Laplace transform

A&S Ref:

7.4.28 (in different notation)

Referenced by:

§7.14(ii)

Permalink:

http://dlmf.nist.gov/7.14.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §7.14(ii), §7.14(ii), §7.14 and Ch.7

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7.14.7 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}C\left(\sqrt{\frac{2t}{\pi }}\right)dt=\frac{{(\sqrt{a^2+1}+a)}^{\frac{1}{2}}}{2a\sqrt{a^2+1}},$$ $\mathrm{\Re }a>0$,

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Symbols:

$C\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Laplace transform

A&S Ref:

7.4.29 in modified form

Referenced by:

§7.14(ii)

Permalink:

http://dlmf.nist.gov/7.14.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §7.14(ii), §7.14(ii), §7.14 and Ch.7

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7.14.8 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}S\left(\sqrt{\frac{2t}{\pi }}\right)dt=\frac{{(\sqrt{a^2+1}-a)}^{\frac{1}{2}}}{2a\sqrt{a^2+1}},$$ $\mathrm{\Re }a>0$.

ⓘ

Symbols:

$S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Laplace transform

A&S Ref:

7.4.30 in modified form

Referenced by:

§7.14(ii)

Permalink:

http://dlmf.nist.gov/7.14.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §7.14(ii), §7.14(ii), §7.14 and Ch.7

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20: 6.7 Integral Representations

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6.7.1 $${\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-at}}{t+b}dt={\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{\mathrm{i}at}}{t+\mathrm{i}b}dt={\mathrm{e}}^{ab}{E}_1\left(ab\right),$$ $a>0$, $b>0$,

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{i}$: imaginary unit and $\int$: integral

A&S Ref:

5.1.28 (modified form of) 5.1.29 (modified form of)

Referenced by:

§6.7(i)

Permalink:

http://dlmf.nist.gov/6.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(i), §6.7 and Ch.6

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6.7.2 $${\mathrm{e}}^x{\int }_0^{\alpha }\frac{{\mathrm{e}}^{-xt}}{1-t}dt=\mathrm{Ei}\left(x\right)-\mathrm{Ei}\left((1-\alpha )x\right),$$ , $x>0$.

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{Ei}\left(x\right)$: exponential integral, $\int$: integral and $x$: real variable

Referenced by:

§6.7(i)

Permalink:

http://dlmf.nist.gov/6.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(i), §6.7 and Ch.6

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6.7.9 $$\mathrm{si}\left(z\right)=-{\int }_0^{\pi /2}{\mathrm{e}}^{-z\cos t}\cos \left(z\sin t\right)dt,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{si}\left(z\right)$: sine integral, $\sin z$: sine function and $z$: complex variable

A&S Ref:

5.2.10

Permalink:

http://dlmf.nist.gov/6.7.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(ii), §6.7 and Ch.6

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6.7.10 $$\mathrm{Ein}\left(z\right)-\mathrm{Cin}\left(z\right)={\int }_0^{\pi /2}{\mathrm{e}}^{-z\cos t}\sin \left(z\sin t\right)dt,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ein}\left(z\right)$: complementary exponential integral, $\cos z$: cosine function, $\mathrm{Cin}\left(z\right)$: cosine integral, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\sin z$: sine function and $z$: complex variable

A&S Ref:

5.2.11 (modified form of)

Permalink:

http://dlmf.nist.gov/6.7.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(ii), §6.7 and Ch.6

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6.7.13 $$\mathrm{f}\left(z\right)={\int }_0^{\mathrm{\infty }}\frac{\sin t}{t+z}dt={\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-zt}}{t^2+1}dt,$$

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\sin z$: sine function, $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

Referenced by:

§6.12(ii), §6.7(iii)

Permalink:

http://dlmf.nist.gov/6.7.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(iii), §6.7 and Ch.6

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