generalized exponential integrals

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21—30 of 118 matching pages

21: Bibliography M

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G. F. Miller (1960) Tables of Generalized Exponential Integrals. NPL Mathematical Tables, Vol. III, Her Majesty’s Stationery Office, London.

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External Links:: MathReview (L. Fox)
Referenced by:: §8.25(v).
Encodings:: BibTeX

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22: 9.13 Generalized Airy Functions

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9.13.27

B_{k}\left(z,p\right)=\int_{\mathscr{I}_{k}}t^{-p}\exp\left(zt-\tfrac{1}{3}t^{% 3}\right)\,\mathrm{d}t,

k=1,2,3

p=0,\pm 1,\pm 2,\ldots

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Symbols:: $B_{\NVar{k}}\left(\NVar{z},\NVar{p}\right)$ : generalized Airy function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $z$ : complex variable, $k$ : index, $p$ : parameter and $\mathscr{I}$ : integration path
Source:: Drazin and Reid (1981, (A20), p. 470)
Permalink:: http://dlmf.nist.gov/9.13.E27
Encodings:: pMML, png, TeX
See also:: Annotations for §9.13(ii), §9.13 and Ch.9

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A_{2}\left(z,0\right)=e^{2\pi i/3}\operatorname{Ai}\left(ze^{2\pi i/3}\right)

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A_{3}\left(z,0\right)=e^{-2\pi i/3}\operatorname{Ai}\left(ze^{-2\pi i/3}\right)

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A_{2}\left(z,p\right)=e^{-2(p-1)\pi i/3}A_{1}\left(ze^{2\pi i/3},p\right)

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A_{3}\left(z,p\right)=e^{2(p-1)\pi i/3}A_{1}\left(ze^{-2\pi i/3},p\right)

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23: 8.21 Generalized Sine and Cosine Integrals

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8.21.1

\operatorname{ci}\left(a,z\right)\pm i\operatorname{si}\left(a,z\right)=e^{\pm% \frac{1}{2}\pi ia}\Gamma\left(a,ze^{\mp\frac{1}{2}\pi i}\right),

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Defines:: $\operatorname{ci}\left(\NVar{a},\NVar{z}\right)$ : generalized cosine integral and $\operatorname{si}\left(\NVar{a},\NVar{z}\right)$ : generalized sine integral
Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $z$ : complex variable and $a$ : parameter
Referenced by:: §8.21(ii)
Permalink:: http://dlmf.nist.gov/8.21.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §8.21(i), §8.21 and Ch.8

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8.21.2

\operatorname{Ci}\left(a,z\right)\pm i\operatorname{Si}\left(a,z\right)=e^{\pm% \frac{1}{2}\pi ia}\gamma\left(a,ze^{\mp\frac{1}{2}\pi i}\right).

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Defines:: $\operatorname{Ci}\left(\NVar{a},\NVar{z}\right)$ : generalized cosine integral and $\operatorname{Si}\left(\NVar{a},\NVar{z}\right)$ : generalized sine integral
Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $z$ : complex variable and $a$ : parameter
Referenced by:: §8.21(ii)
Permalink:: http://dlmf.nist.gov/8.21.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §8.21(i), §8.21 and Ch.8

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24: 7.16 Generalized Error Functions

… ►Generalizations of the error function and Dawson’s integral are

\int_{0}^{x}e^{-t^{p}}\,\mathrm{d}t

and

\int_{0}^{x}e^{t^{p}}\,\mathrm{d}t

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25: 6 Exponential, Logarithmic, Sine, and
Cosine Integrals

Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals

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26: 4.7 Derivatives and Differential Equations

… ►For a nonvanishing analytic function

f(z)

, the general solution of the differential equation … ►

§4.7(ii) Exponentials and Powers

… ►When

a^{z}

is a general power,

\ln a

is replaced by the branch of

\operatorname{Ln}a

used in constructing

a^{z}

. … ►The general solution of the differential equation … ►The general solution of the differential equation …

27: 5.9 Integral Representations

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5.9.10

\operatorname{Ln}\Gamma\left(z\right)=\left(z-\tfrac{1}{2}\right)\ln z-z+% \tfrac{1}{2}\ln\left(2\pi\right)+2\int_{0}^{\infty}\frac{\operatorname{arctan}% \left(t/z\right)}{e^{2\pi t}-1}\,\mathrm{d}t,

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\operatorname{Ln}\NVar{z}$ : general logarithm function, $\ln\NVar{z}$ : principal branch of logarithm function and $z$ : complex variable
A&S Ref:: 6.1.50
Referenced by:: (5.9.10_1), §5.9(i), §5.9(ii), Erratum (V1.0.10) for Equations (5.9.10), (5.9.11), (5.10.1), (5.11.1), (5.11.8)
Permalink:: http://dlmf.nist.gov/5.9.E10
Encodings:: pMML, png, TeX
Addition (effective with 1.0.10):: To increase the region of validity of this equation, the logarithm of the gamma function that appears on its left-hand side has been changed to $\operatorname{Ln}\Gamma\left(z\right)$ , where $\operatorname{Ln}$ is the general logarithm. Originally $\ln\Gamma\left(z\right)$ was used, where $\ln$ is the principal branch of the logarithm.
Suggested 2015-02-13 by Philippe Spindel
See also:: Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

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5.9.10_1

\operatorname{Ln}\Gamma\left(z\right)=\left(z-\tfrac{1}{2}\right)\ln z-z+% \tfrac{1}{2}\ln\left(2\pi\right)-\frac{z}{\pi}\int_{0}^{\infty}\frac{\ln\left(% 1-{\mathrm{e}}^{-2\pi t}\right)}{t^{2}+z^{2}}\,\mathrm{d}t,

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{Ln}\NVar{z}$ : general logarithm function, $\ln\NVar{z}$ : principal branch of logarithm function and $z$ : complex variable
Proof sketch:: Can be obtained from (5.9.10) via integration by parts.
Referenced by:: §5.9(i), Erratum (V1.1.7) for Additions
Permalink:: http://dlmf.nist.gov/5.9.E10_1
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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5.9.10_2

\operatorname{Ln}\Gamma\left(z\right)=\left(z-\tfrac{1}{2}\right)\ln z-z+% \tfrac{1}{2}\ln\left(2\pi\right)+\int_{0}^{\infty}{\mathrm{e}}^{-zt}\left(% \frac{1}{{\mathrm{e}}^{t}-1}-\frac{1}{t}+\frac{1}{2}\right)\frac{\,\mathrm{d}t% }{t},

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{Ln}\NVar{z}$ : general logarithm function, $\ln\NVar{z}$ : 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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28: 35.6 Confluent Hypergeometric Functions of Matrix Argument

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35.6.4

{{}_{1}F_{1}}\left({a\atop b};\mathbf{T}\right)=\frac{1}{\mathrm{B}_{m}\left(a% ,b-a\right)}\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{I}}\operatorname{% etr}\left(\mathbf{T}\mathbf{X}\right)\left|\mathbf{X}\right|^{a-\frac{1}{2}(m+% 1)}\left|\mathbf{I}-\mathbf{X}\right|^{b-a-\frac{1}{2}(m+1)}\,\mathrm{d}{% \mathbf{X}},

\Re\left(a\right),\Re\left(b-a\right)>\frac{1}{2}(m-1)

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Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $\operatorname{etr}\left(\NVar{\mathbf{A}}\right)$ : exponential of trace, $\int$ : integral, $\mathrm{B}_{\NVar{m}}\left(\NVar{a},\NVar{b}\right)$ : multivariate beta function, $\Re$ : real part, $a$ : complex variable, $\mathbf{I}$ : $m\times m$ identity matrix, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $\mathbf{X}$ : real symmetric $m\times m$ matrix, $\left|\NVar{\mathbf{X}}\right|$ : determinant of $\NVar{\mathbf{X}}$ , $\,\mathrm{d}$ : differential of matrix, $b$ : complex variable, $<$ : less-than for matrices, $m$ : positive integer and $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros
Permalink:: http://dlmf.nist.gov/35.6.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §35.6(ii), §35.6 and Ch.35

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35.6.5

\int_{\boldsymbol{\Omega}}\operatorname{etr}\left(-\mathbf{T}\mathbf{X}\right)% \left|\mathbf{X}\right|^{b-\frac{1}{2}(m+1)}{{}_{1}F_{1}}\left({a\atop b};% \mathbf{S}\mathbf{X}\right)\,\mathrm{d}{\mathbf{X}}=\Gamma_{m}\left(b\right)% \left|\mathbf{I}-\mathbf{S}\mathbf{T}^{-1}\right|^{-a}\left|\mathbf{T}\right|^% {-b},

\mathbf{T}>\mathbf{S}

\Re\left(b\right)>\frac{1}{2}(m-1)

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Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $\operatorname{etr}\left(\NVar{\mathbf{A}}\right)$ : exponential of trace, $\int$ : integral, $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function, $\Re$ : real part, $a$ : complex variable, $\mathbf{I}$ : $m\times m$ identity matrix, $\mathbf{S}$ : real symmetric $m\times m$ matrix, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $\mathbf{X}$ : real symmetric $m\times m$ matrix, $\left|\NVar{\mathbf{X}}\right|$ : determinant of $\NVar{\mathbf{X}}$ , $\,\mathrm{d}$ : differential of matrix, ${\boldsymbol{\Omega}}$ : space of positive-definite real symmetric $m\times m$ matrices, $b$ : complex variable, $>$ : greater-than for matrices and $m$ : positive integer
Referenced by:: §35.6(ii)
Permalink:: http://dlmf.nist.gov/35.6.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §35.6(ii), §35.6 and Ch.35

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29: 16.5 Integral Representations and Integrals

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16.5.3

{{}_{p+1}F_{q}}\left({a_{0},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)=\frac% {1}{\Gamma\left(a_{0}\right)}\int_{0}^{\infty}{\mathrm{e}}^{-t}t^{a_{0}-1}{{}_% {p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\cdots,b_{q}};zt\right)\,\mathrm{% d}t,

\Re z<1

\Re a_{0}>0

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part, $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Keywords:: Laplace transform, Mellin transform
Referenced by:: §16.5
Permalink:: http://dlmf.nist.gov/16.5.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §16.5 and Ch.16

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16.5.4

{{}_{p}F_{q+1}}\left({a_{1},\dots,a_{p}\atop b_{0},\dots,b_{q}};z\right)=\frac% {\Gamma\left(b_{0}\right)}{2\pi\mathrm{i}}\int_{c-\mathrm{i}\infty}^{c+\mathrm% {i}\infty}{\mathrm{e}}^{t}t^{-b_{0}}{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b% _{1},\dots,b_{q}};\frac{z}{t}\right)\,\mathrm{d}t,

c>0

\Re b_{0}>0

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part, $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Keywords:: Mellin transform
Referenced by:: §16.5, §16.5
Permalink:: http://dlmf.nist.gov/16.5.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §16.5 and Ch.16

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30: 35.8 Generalized Hypergeometric Functions of Matrix Argument

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35.8.12

{\int_{\boldsymbol{\Omega}}\operatorname{etr}\left(-\mathbf{T}\mathbf{X}\right% )\left|\mathbf{X}\right|^{\gamma-\frac{1}{2}(m+1)}\*{{}_{p}F_{q}}\left({a_{1},% \dots,a_{p}\atop b_{1},\dots,b_{q}};-\mathbf{X}\right)\,\mathrm{d}{\mathbf{X}}% }=\Gamma_{m}\left(\gamma\right)\left|\mathbf{T}\right|^{-\gamma}{{}_{p+1}F_{q}% }\left({a_{1},\dots,a_{p},\gamma\atop b_{1},\dots,b_{q}};-\mathbf{T}^{-1}% \right),

\Re\left(\gamma\right)>\frac{1}{2}(m-1)

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Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $\operatorname{etr}\left(\NVar{\mathbf{A}}\right)$ : exponential of trace, $\int$ : integral, $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function, $\Re$ : real part, $a$ : complex variable, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $\mathbf{X}$ : real symmetric $m\times m$ matrix, $\left|\NVar{\mathbf{X}}\right|$ : determinant of $\NVar{\mathbf{X}}$ , $\,\mathrm{d}$ : differential of matrix, ${\boldsymbol{\Omega}}$ : space of positive-definite real symmetric $m\times m$ matrices, $b$ : complex variable, $p$ : nonnegative integer, $q$ : nonnegative integer and $m$ : positive integer
Permalink:: http://dlmf.nist.gov/35.8.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §35.8(iv), §35.8(iv), §35.8 and Ch.35

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