Cauchy determinant

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31—40 of 68 matching pages

31: 1.13 Differential Equations

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1.13.4

\mathscr{W}\left\{w_{1}(z),w_{2}(z)\right\}=\det\begin{bmatrix}w_{1}(z)&w_{2}(% z)\\ w_{1}^{\prime}(z)&w_{2}^{\prime}(z)\end{bmatrix}=w_{1}(z)w_{2}^{\prime}(z)-w_{% 2}(z)w_{1}^{\prime}(z).

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Symbols:: $\mathscr{W}$ : Wronskian, $\det$ : determinant, $z$ : variable, $w_{1}(z)$ : solution and $w_{2}(z)$ : solution
Referenced by:: §1.13(i), Erratum (V1.0.25) for Section 1.13
Permalink:: http://dlmf.nist.gov/1.13.E4
Encodings:: pMML, png, TeX
Addition (effective with 1.0.25):: The determinant form of the two-argument Wronskian was added as an equality to this equation.
See also:: Annotations for §1.13(i), §1.13(i), §1.13 and Ch.1

►(More generally

\mathscr{W}\left\{w_{1}(z),\ldots,w_{n}(z)\right\}=\det\left[w_{k}^{(j-1)}(z)\right]

, where

1\leq j,k\leq n

.) …

33: 3.2 Linear Algebra

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3.2.19

p_{n}(\lambda)=\det[\lambda\mathbf{I}-\mathbf{A}]

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Symbols:: $\det$ : determinant and $\lambda$ : eigenvalue
Permalink:: http://dlmf.nist.gov/3.2.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §3.2(iv), §3.2 and Ch.3

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… ►For an extension to integrals with Cauchy principal values see Elliott (1998). … ►and Cauchy’s theorem, we have … ►These problems can be brought within the scope of §2.4 by means of Cauchy’s integral formula … ►By allowing the contour in Cauchy’s formula to expand, we find that …

35: 9.10 Integrals

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9.10.19

\operatorname{Bi}\left(x\right)=\frac{3x^{5/4}e^{(2/3)x^{3/2}}}{2\pi}\*\pvint_% {0}^{\infty}\frac{t^{-3/4}e^{-(2/3)t^{3/2}}\operatorname{Ai}\left(t\right)}{x^% {3/2}-t^{3/2}}\,\mathrm{d}t,

x>0

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy 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, $\pvint_{\NVar{a}}^{\NVar{b}}$ : Cauchy principal value and $x$ : real variable
Proof sketch:: To prove this equation, combine (9.2.10) with (9.10.18).
Referenced by:: §9.5(i), Erratum (V1.0.10) for Equation (9.10.19)
Permalink:: http://dlmf.nist.gov/9.10.E19
Encodings:: pMML, png, TeX
Errata (effective with 1.0.10):: The original and incorrect version,
$\operatorname{Bi}\left(x\right)=\frac{x^{5/4}e^{(2/3)x^{3/2}}}{2^{5/2}\pi}\*% \pvint_{0}^{\infty}\frac{t^{-1/2}e^{-(2/3)t^{3/2}}\operatorname{Ai}\left(t% \right)}{x^{3/2}-t^{3/2}}\,\mathrm{d}t$
taken from Schulten et al. (1979), has been corrected.
Reported 2015-03-20 by Walter Gautschi
See also:: Annotations for §9.10(vii), §9.10 and Ch.9

►where the last integral is a Cauchy principal value (§1.4(v)). …

36: Mathematical Introduction

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$\mathbb{C}$	complex plane (excluding infinity).
…
$\det$	determinant.
…

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37: Bibliography V

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R. Vein and P. Dale (1999) Determinants and Their Applications in Mathematical Physics. Applied Mathematical Sciences, Vol. 134, Springer-Verlag, New York.

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External Links:: ISBN 0-387-98558-1, MathReview (Jorma Kaarlo Merikoski), ZentralBlatt
Referenced by:: §1.3(ii).
Encodings:: BibTeX

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38: 35.7 Gaussian Hypergeometric Function of Matrix Argument

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35.7.4

\lim_{c\to\infty}{{}_{2}F_{1}}\left({a,b\atop c};\mathbf{I}-c\mathbf{T}^{-1}% \right)=\left|\mathbf{T}\right|^{b}\Psi\left(b;b-a+\tfrac{1}{2}(m+1);\mathbf{T% }\right).

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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, $\Psi\left(\NVar{a};\NVar{b};\NVar{\mathbf{T}}\right)$ : confluent hypergeometric function of matrix argument (second kind), $a$ : complex variable, $\mathbf{I}$ : $m\times m$ identity matrix, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $\left|\NVar{\mathbf{X}}\right|$ : determinant of $\NVar{\mathbf{X}}$ , $b$ : complex variable, $c$ : complex variable and $m$ : positive integer
Permalink:: http://dlmf.nist.gov/35.7.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35

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35.7.5

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

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

\boldsymbol{{0}}<\mathbf{T}<\mathbf{I}

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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, $\int$ : 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, $c$ : complex variable, $m$ : positive integer and $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros
Referenced by:: §35.7(iv)
Permalink:: http://dlmf.nist.gov/35.7.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35

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35.7.6

{{}_{2}F_{1}}\left({a,b\atop c};\mathbf{T}\right)=\left|\mathbf{I}-\mathbf{T}% \right|^{c-a-b}{{}_{2}F_{1}}\left({c-a,c-b\atop c};\mathbf{T}\right)=\left|% \mathbf{I}-\mathbf{T}\right|^{-a}{{}_{2}F_{1}}\left({a,c-b\atop c};-\mathbf{T}% (\mathbf{I}-\mathbf{T})^{-1}\right)=\left|\mathbf{I}-\mathbf{T}\right|^{-b}{{}% _{2}F_{1}}\left({c-a,b\atop c};-\mathbf{T}(\mathbf{I}-\mathbf{T})^{-1}\right).

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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, $a$ : complex variable, $\mathbf{I}$ : $m\times m$ identity matrix, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $\left|\NVar{\mathbf{X}}\right|$ : determinant of $\NVar{\mathbf{X}}$ , $b$ : complex variable and $c$ : complex variable
Permalink:: http://dlmf.nist.gov/35.7.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35

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39: 35.8 Generalized Hypergeometric Functions of Matrix Argument

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35.8.3

{{}_{2}F_{1}}\left({a,b\atop b};\mathbf{T}\right)={{}_{1}F_{0}}\left({a\atop-}% ;\mathbf{T}\right)=\left|\mathbf{I}-\mathbf{T}\right|^{-a},

\boldsymbol{{0}}<\mathbf{T}<\mathbf{I}

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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, $a$ : complex variable, $\mathbf{I}$ : $m\times m$ identity matrix, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $\left|\NVar{\mathbf{X}}\right|$ : determinant of $\NVar{\mathbf{X}}$ , $b$ : complex variable, $<$ : less-than for matrices and $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros
Permalink:: http://dlmf.nist.gov/35.8.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §35.8(ii), §35.8 and Ch.35

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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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35.8.13

\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{I}}\left|\mathbf{X}\right|^{a% _{1}-\frac{1}{2}(m+1)}{\left|\mathbf{I}-\mathbf{X}\right|}^{b_{1}-a_{1}-\frac{% 1}{2}(m+1)}\*{{}_{p}F_{q}}\left({a_{2},\dots,a_{p+1}\atop b_{2},\dots,b_{q+1}}% ;\mathbf{T}\mathbf{X}\right)\,\mathrm{d}{\mathbf{X}}=\frac{1}{\mathrm{B}_{m}% \left(b_{1}-a_{1},a_{1}\right)}{{}_{p+1}F_{q+1}}\left({a_{1},\dots,a_{p+1}% \atop b_{1},\dots,b_{q+1}};\mathbf{T}\right),

\Re\left(b_{1}-a_{1}\right),\Re\left(a_{1}\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, $\int$ : 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, $p$ : nonnegative integer, $q$ : nonnegative integer, $m$ : positive integer and $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros
Permalink:: http://dlmf.nist.gov/35.8.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §35.8(iv), §35.8(iv), §35.8 and Ch.35

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40: Bibliography H

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P. Henrici (1986) Applied and Computational Complex Analysis. Vol. 3: Discrete Fourier Analysis—Cauchy Integrals—Construction of Conformal Maps—Univalent Functions. Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons Inc.], New York.

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Notes:: Reprinted in 1993.
External Links:: ISBN 0-471-08703-3; 0-471-58986-1, MathReview (A. E. Heins), ZentralBlatt
Referenced by:: §1.14(v), Ch.3.
Encodings:: BibTeX

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