definite integral

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21: Errata

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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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22: 35.2 Laplace Transform

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35.2.1 $$g(𝐙)={\int }_{𝛀}\mathrm{etr}\left(-𝐙𝐗\right)f(𝐗)d𝐗,$$

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Defines:

$g$: Laplace transformation of $f$ (locally)

Symbols:

$\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, $𝐗$: real symmetric $m\times m$ matrix, $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices, $𝐙$: complex symmetric $m\times m$ matrix and $f(𝐗)$: complex-valued function of matrix argument

Keywords:

Laplace transform

Referenced by:

§35.2

Permalink:

http://dlmf.nist.gov/35.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §35.2, §35.2 and Ch.35

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23: 35.5 Bessel Functions of Matrix Argument

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35.5.3 $$B_{\nu }\left(𝐓\right)={\int }_{𝛀}\mathrm{etr}\left(-(𝐓𝐗+{𝐗}^{-1})\right){\left|𝐗\right|}^{\nu -\frac{1}{2}(m+1)}d𝐗,$$ $\nu \in ℂ$, $𝐓\in 𝛀$.

ⓘ

Defines:

$B_{\nu }\left(𝐓\right)$: Bessel function of matrix argument (second kind)

Symbols:

$ℂ$: complex plane, $\in$: element of, $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, $𝐓$: 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 and $m$: positive integer

Referenced by:

§35.9

Permalink:

http://dlmf.nist.gov/35.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §35.5(i), §35.5 and Ch.35

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35.5.4 $${\int }_{𝛀}\mathrm{etr}\left(-𝐓𝐗\right){\left|𝐗\right|}^{\nu }{A}_{\nu }\left(𝐒𝐗\right)d𝐗=\mathrm{etr}\left(-𝐒{𝐓}^{-1}\right){\left|𝐓\right|}^{-\nu -\frac{1}{2}(m+1)},$$ $𝐒\in 𝓢$, $𝐓\in 𝛀$; $\mathrm{\Re }\left(\nu \right)>-1$.

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Symbols:

$A_{\nu }\left(𝐓\right)$: Bessel function of matrix argument (first kind), $\in$: element of, $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, $\mathrm{\Re }$: real part, $𝓢$: space of real symmetric $m\times m$ matrices, $𝐒$: real symmetric $m\times m$ matrix, $𝐓$: 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 and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §35.5(ii), §35.5 and Ch.35

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35.5.5 ${\nu }_j\in ℂ$, $\mathrm{\Re }\left({\nu }_j\right)>-1$, $j=1,2$; ${𝐒}_1,{𝐒}_2\in 𝓢$; $𝐓\in 𝛀$.

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Symbols:

$A_{\nu }\left(𝐓\right)$: Bessel function of matrix argument (first kind), $ℂ$: complex plane, $\in$: element of, $\int$: integral, $\mathrm{\Re }$: real part, $𝓢$: space of real symmetric $m\times m$ matrices, $𝐒$: real symmetric $m\times m$ matrix, $𝐓$: 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, : less-than for matrices, $j$: nonnegative integer, $m$: positive integer and $𝟎$: $m\times m$ matrix of zeros

Permalink:

http://dlmf.nist.gov/35.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §35.5(ii), §35.5 and Ch.35

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35.5.7 $${\int }_{𝛀}{A}_{{\nu }_1}\left(𝐓𝐗\right)B_{-{\nu }_2}\left(𝐒𝐗\right){\left|𝐗\right|}^{{\nu }_1}d𝐗=\frac{1}{A_{{\nu }_1+{\nu }_2}\left(𝟎\right)}{\left|𝐒\right|}^{{\nu }_2}{\left|𝐓+𝐒\right|}^{-({\nu }_1+{\nu }_2+\frac{1}{2}(m+1))},$$ $\mathrm{\Re }\left({\nu }_1+{\nu }_2\right)>-1$; $𝐒,𝐓\in 𝛀$.

ⓘ

Symbols:

$A_{\nu }\left(𝐓\right)$: Bessel function of matrix argument (first kind), $B_{\nu }\left(𝐓\right)$: Bessel function of matrix argument (second kind), $\in$: element of, $\int$: integral, $\mathrm{\Re }$: real part, $𝐒$: real symmetric $m\times m$ matrix, $𝐓$: 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, $m$: positive integer and $𝟎$: $m\times m$ matrix of zeros

Permalink:

http://dlmf.nist.gov/35.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §35.5(ii), §35.5 and Ch.35

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24: 35.3 Multivariate Gamma and Beta Functions

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35.3.1 $${\mathrm{\Gamma }}_m\left(a\right)={\int }_{𝛀}\mathrm{etr}\left(-𝐗\right){\left|𝐗\right|}^{a-\frac{1}{2}(m+1)}d𝐗,$$ $\mathrm{\Re }\left(a\right)>\frac{1}{2}(m-1)$.

ⓘ

Symbols:

$\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐗$: 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 and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.3.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §35.3(i), §35.3 and Ch.35

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35.3.2 $${\mathrm{\Gamma }}_m\left(s_1,\mathrm{\dots },s_m\right)={\int }_{𝛀}\mathrm{etr}\left(-𝐗\right){\left|𝐗\right|}^{s_m-\frac{1}{2}(m+1)}\prod _{j=1}^{m-1}{|{(𝐗)}_j|}^{s_j-s_{j+1}}d𝐗,$$ $s_j\in ℂ$, $\mathrm{\Re }\left(s_j\right)>\frac{1}{2}(j-1)$, $j=1,\mathrm{\dots },m$.

ⓘ

Symbols:

$ℂ$: complex plane, $\in$: element of, $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $|{(𝐗)}_j|$: $j$th principal minor of $𝐗$ , $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices, $j$: nonnegative integer and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.3.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §35.3(i), §35.3 and Ch.35

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35.3.8 $${\mathrm{B}}_m(a,b)={\int }_{𝛀}{\left|𝐗\right|}^{a-\frac{1}{2}(m+1)}{\left|𝐈+𝐗\right|}^{-(a+b)}d𝐗,$$ $\mathrm{\Re }\left(a\right),\mathrm{\Re }\left(b\right)>\frac{1}{2}(m-1)$.

ⓘ

Symbols:

$\int$: integral, ${\mathrm{B}}_m(a,b)$: multivariate beta function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐈$: $m\times m$ identity 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 and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.3.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §35.3(ii), §35.3 and Ch.35

25: 35.6 Confluent Hypergeometric Functions of Matrix Argument

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35.6.2 $$\mathrm{\Psi }(a;b;𝐓)=\frac{1}{{\mathrm{\Gamma }}_m\left(a\right)}{\int }_{𝛀}\mathrm{etr}\left(-𝐓𝐗\right){\left|𝐗\right|}^{a-\frac{1}{2}(m+1)}{\left|𝐈+𝐗\right|}^{b-a-\frac{1}{2}(m+1)}d𝐗,$$ $\mathrm{\Re }\left(a\right)>\frac{1}{2}(m-1)$, $𝐓\in 𝛀$.

ⓘ

Defines:

$\mathrm{\Psi }(a;b;𝐓)$: confluent hypergeometric function of matrix argument (second kind)

Symbols:

$\in$: element of, $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐓$: 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 and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §35.6(i), §35.6 and Ch.35

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35.6.5 $${\int }_{𝛀}\mathrm{etr}\left(-𝐓𝐗\right){\left|𝐗\right|}^{b-\frac{1}{2}(m+1)}{}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a}{b};𝐒𝐗)d𝐗={\mathrm{\Gamma }}_m\left(b\right){\left|𝐈-𝐒{𝐓}^{-1}\right|}^{-a}{\left|𝐓\right|}^{-b},$$ $𝐓>𝐒$, $\mathrm{\Re }\left(b\right)>\frac{1}{2}(m-1)$.

ⓘ

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, $𝐈$: $m\times m$ identity matrix, $𝐒$: real symmetric $m\times m$ matrix, $𝐓$: 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, $>$: 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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35.6.8 $${\int }_{𝛀}{\left|𝐓\right|}^{c-\frac{1}{2}(m+1)}\mathrm{\Psi }(a;b;𝐓)d𝐓=\frac{{\mathrm{\Gamma }}_m\left(c\right){\mathrm{\Gamma }}_m\left(a-c\right){\mathrm{\Gamma }}_m\left(c-b+\frac{1}{2}(m+1)\right)}{{\mathrm{\Gamma }}_m\left(a\right){\mathrm{\Gamma }}_m\left(a-b+\frac{1}{2}(m+1)\right)},$$ $\mathrm{\Re }\left(a\right)>\mathrm{\Re }\left(c\right)+\frac{1}{2}(m-1)>m-1$, $\mathrm{\Re }\left(c-b\right)>-1$.

ⓘ

Symbols:

$\mathrm{\Psi }(a;b;𝐓)$: confluent hypergeometric function of matrix argument (second kind), $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐓$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices, $b$: complex variable, $c$: complex variable and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §35.6(ii), §35.6 and Ch.35

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26: 35.4 Partitions and Zonal Polynomials

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35.4.8 $${\int }_{𝛀}\mathrm{etr}\left(-𝐓𝐗\right){\left|𝐗\right|}^{a-\frac{1}{2}(m+1)}{Z}_{\kappa }\left(𝐗\right)d𝐗={\mathrm{\Gamma }}_m\left(a+\kappa \right){\left|𝐓\right|}^{-a}{Z}_{\kappa }\left({𝐓}^{-1}\right),$$

ⓘ

Symbols:

$\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $a$: 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, $m$: positive integer and $\kappa$: vector of nonnegative integers

Permalink:

http://dlmf.nist.gov/35.4.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35

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27: 19.31 Probability Distributions

§19.31 Probability Distributions

► $R_G(x,y,z)$ and $R_F(x,y,z)$ occur as the expectation values, relative to a normal probability distribution in ${ℝ}^2$ or ${ℝ}^3$, of the square root or reciprocal square root of a quadratic form. More generally, let $𝐀$ ($=[a_{r,s}]$) and $𝐁$ ($=[b_{r,s}]$) be real positive-definite matrices with $n$ rows and $n$ columns, and let ${\lambda }_1,\mathrm{\dots },{\lambda }_n$ be the eigenvalues of $𝐀{𝐁}^{-1}$. … ►

19.31.2 $${\int }_{{ℝ}^n}{({𝐱}^{\mathrm{T}}𝐀𝐱)}^{\mu }\exp \left(-{𝐱}^{\mathrm{T}}𝐁𝐱\right)d{x}_1\mathrm{\cdots }d{x}_n=\frac{{\pi }^{n/2}\mathrm{\Gamma }\left(\mu +\frac{1}{2}n\right)}{\sqrt{det𝐁}\mathrm{\Gamma }\left(\frac{1}{2}n\right)}{R}_{\mu }(\frac{1}{2},\mathrm{\dots },\frac{1}{2};{\lambda }_1,\mathrm{\dots },{\lambda }_n),$$ $\mu >-\frac{1}{2}n$.

ⓘ

Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $det$: determinant, $dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $ℝ$: real line, ${𝐀}^{\mathrm{T}}$: transpose of matrix, $n$: nonnegative integer and ${\lambda }_n$: eigenvalues

Referenced by:

§19.31

Permalink:

http://dlmf.nist.gov/19.31.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.31 and Ch.19

►§19.16(iii) shows that for $n=3$ the incomplete cases of $R_F$ and $R_G$ occur when $\mu =-1/2$ and $\mu =1/2$, respectively, while their complete cases occur when $n=2$. …

28: 35.8 Generalized Hypergeometric Functions of Matrix Argument

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

ⓘ

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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29: 18.34 Bessel Polynomials

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18.34.5_5 $$\frac{2^{1-a}}{\mathrm{\Gamma }\left(1-a\right)}{\int }_0^{\mathrm{\infty }}{y}_n(x;a)y_m(x;a)x^{a-2}{\mathrm{e}}^{-2x^{-1}}dx=\frac{1-a}{1-a-2n}\frac{n!}{{\left(2-a-n\right)}_n}{\delta }_{n,m},$$ $m,n=0,1,\mathrm{\dots },N=\lceil -(1+a)/2\rceil$.

ⓘ

Symbols:

$y_n(x;a)$: Bessel polynomial, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\delta }_{j,k}$: Kronecker delta, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\lceil x\rceil$: ceiling of $x$ , $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $N$: positive integer, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Proved:

Lesky (1996, p.183, case a.3))(proved)

Sources:

Romanovski (1929, Type V); Koekoek et al. (2010, (9.13.2))

Referenced by:

(18.34.7_3), §18.34(ii), §18.34(iii), Erratum (V1.2.0) Section 18.34

Permalink:

http://dlmf.nist.gov/18.34.E5_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.34(ii), §18.34 and Ch.18

►The full system satisfies orthogonality with respect to a (not positive definite) moment functional; see Evans et al. (1993, (2.7)) for the simple expression of the moments ${\mu }_n$. … ►

18.34.6 $$\frac{1}{2\pi \mathrm{i}}{\int }_{|z|=1}{z}^{a-2}{y}_n(z;a)y_m(z;a){\mathrm{e}}^{-2/z}dz=\frac{{(-1)}^{n+a-1}n!{\mathrm{\hspace{0.17em}2}}^{a-1}}{(n+a-2)!(2n+a-1)}{\delta }_{n,m},$$ $a=1,2,\mathrm{\dots }$,

ⓘ

Symbols:

$y_n(x;a)$: Bessel polynomial, ${\delta }_{j,k}$: Kronecker delta, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\int$: integral, $z$: complex variable, $m$: nonnegative integer and $n$: nonnegative integer

Proved:

Grosswald (1978, p.32, Corollary 16)(proved)

Referenced by:

§18.34(ii)

Permalink:

http://dlmf.nist.gov/18.34.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §18.34(ii), §18.34 and Ch.18

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18.34.7_3 $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\varphi }_n(x;\lambda ){\varphi }_m(x;\lambda )dx=\frac{\mathrm{\Gamma }\left(2\lambda -n\right)}{(2\lambda -2n-1)n!}{\delta }_{n,m},$$ $m,n=0,1,\mathrm{\dots },N=\lceil \lambda -\frac{3}{2}\rceil$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\delta }_{j,k}$: Kronecker delta, $\lceil x\rceil$: ceiling of $x$ , $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $N$: positive integer, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Proof sketch:

This follows by straightforward computation from (18.34.5_5) and (18.34.7_1).

Referenced by:

§18.34(iii), (18.39.17), §18.39(i), Erratum (V1.2.0) Section 18.34

Permalink:

http://dlmf.nist.gov/18.34.E7_3

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.34(iii), §18.34 and Ch.18

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30: 1.5 Calculus of Two or More Variables

… ►and the second order term in (1.5.18) is positive definite (negative definite), that is, … ►

Finite Integrals

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Infinite Integrals

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Double Integrals

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Triple Integrals

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