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11: 3.2 Linear Algebra

… ►In the case that the orthogonality condition is replaced by $𝐒$-orthogonality, that is, ${𝐯}_j^{\mathrm{T}}𝐒{𝐯}_k={\delta }_{j,k}$, $j,k=1,2,\mathrm{\dots },n$, for some positive definite matrix $𝐒$ with Cholesky decomposition $𝐒={𝐋}^{\mathrm{T}}𝐋$, then the details change as follows. …

12: 35.8 Generalized Hypergeometric Functions of Matrix Argument

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35.8.12 $${\int }_{𝛀}\mathrm{etr}\left(-𝐓𝐗\right){\left|𝐗\right|}^{\gamma -\frac{1}{2}(m+1)}{}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};-𝐗)d𝐗={\mathrm{\Gamma }}_m\left(\gamma \right){\left|𝐓\right|}^{-\gamma }{}_{p+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p,\gamma }{b_1,\mathrm{\dots },b_q};-{𝐓}^{-1}),$$ $\mathrm{\Re }\left(\gamma \right)>\frac{1}{2}(m-1)$.

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐓$: real symmetric $m\times m$ matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $𝛀$: 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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13: 3.11 Approximation Techniques

… ►The matrix is symmetric and positive definite, but the system is ill-conditioned when $n$ is large because the lower rows of the matrix are approximately proportional to one another. …

14: Bibliography T

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Some definite integrals and Fourier series for Jacobian elliptic functions. Z. Angew. Math. Mech. 49, pp. 95–96.

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External Links:

ISSN 0044-2267, Document, MathReview (B. C. Carlson), ZentralBlatt

Referenced by:

§22.11.

Encodings:

BibTeX

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Parabolic cylinder functions $U(n,x)$ for natural $n$ and positive $x$ . Comput. Phys. Commun. 69, pp. 415–419.

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Notes:

Double-precision Fortran, minimum accuracy 14S, maximum accuracy 21D.

External Links:

Document, Code

Referenced by:

4th item.

Encodings:

BibTeX

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Simplified calculation of $\mathrm{Ei}(x)$ for positive arguments, and a short table of $\mathrm{Shi}(x)$ . Math. Comp. 22 (102), pp. 448–449.

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External Links:

FullText, Document, ZentralBlatt

Referenced by:

§6.18(i).

Encodings:

BibTeX

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15: Guide to Searching the DLMF

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Table 1: Query Examples

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Query	Matching records contain
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int_$^$ sin	any definite integral of sin
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adj, prec/n, and near/n, where n is any positive natural number.

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Table 2: Wildcard Examples

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Query	What it stands for
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int_$^$ sin	any definite integral of sin.

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16: 9.11 Products

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§9.11(iii) Integral Representations

… ►For $\int z^n{w}_1{w}_{2}dz$, $\int z^n{w}_1{w}_2^{\prime }dz$, $\int z^n{w}_1^{\prime }{w}_2^{\prime }dz$, where $n$ is any positive integer, see Albright (1977). … ►

§9.11(v) Definite Integrals

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9.11.19 $${\int }_0^{\mathrm{\infty }}\frac{dt}{{\mathrm{Ai}}^2\left(t\right)+{\mathrm{Bi}}^2\left(t\right)}={\int }_0^{\mathrm{\infty }}\frac{tdt}{\mathrm{Ai}_{}^{\prime }{}_{}{}^2\left(t\right)+\mathrm{Bi}_{}^{\prime }{}_{}{}^2\left(t\right)}=\frac{{\pi }^2}{6}.$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $M\left(z\right)$: Airy modulus function, $N\left(z\right)$: Airy modulus function, $\varphi \left(z\right)$: Airy phase function, $\theta \left(z\right)$: Airy phase function and $x$: real variable

Proof sketch:

Extend the definitions of §9.8(i) to positive values of $x$, obtain the indefinite integrals of $1/M^2\left(x\right)$ and $x/N^2\left(x\right)$ via the first two of (9.8.14), then combine the values of $\theta \left(0\right)$ and $\varphi \left(0\right)$ given in §9.8(i) with $\theta \left(+\mathrm{\infty }\right)=\varphi \left(+\mathrm{\infty }\right)=0$ obtained from (9.8.4), (9.8.8), and §9.7(ii). (Communicated by M.E. Muldoon.)

Permalink:

http://dlmf.nist.gov/9.11.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §9.11(v), §9.11 and Ch.9

►For further definite integrals see Prudnikov et al. (1990, §1.8.2), Laurenzi (1993), Reid (1995, 1997a, 1997b), and Vallée and Soares (2010, Chapters 3, 4).

17: Errata

… ►For some classical polynomials we give some positive sums and discriminants. … ►

33.14.15 $${\int }_0^{\mathrm{\infty }}{\varphi }_{m,\mathrm{\ell }}(r){\varphi }_{n,\mathrm{\ell }}(r)dr={\delta }_{m,n}$$

The definite integral, originally written as ${\int }_0^{\mathrm{\infty }}{\varphi }_{n,\mathrm{\ell }}^2(r)dr=1$, was clarified and rewritten as an orthogonality relation. This follows from (33.14.14) by combining it with Dunkl (2003, Theorem 2.2).

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Originally the function $\chi$ was presented with argument given by a positive integer $n$. It has now been clarified to be valid for argument given by a positive real number $x$.

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The condition for (1.2.2), (1.2.4), and (1.2.5) was corrected. These equations are true only if $n$ is a positive integer. Previously $n$ was allowed to be zero.

Reported 2011-08-10 by Michael Somos.

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The condition for the validity of (8.17.5) is that $m$ and $n$ are positive integers and . Previously, no conditions were stated.

Reported 2011-03-23 by Stephen Bourn.

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18: Bibliography W

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On the points of inflection of Bessel functions of positive order. II. Canad. J. Math. 43 (3), pp. 628–651.

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External Links:

ISSN 0008-414X, MathReview (Evangelos Ifantis), ZentralBlatt

Referenced by:

§10.21(iv).

Encodings:

BibTeX

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On uniform asymptotic expansion of definite integrals. J. Approximation Theory 7 (1), pp. 76–86.

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External Links:

Document, MathReview (L. Berg), ZentralBlatt

Referenced by:

§8.11(v).

Encodings:

BibTeX

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