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31: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►

1.3.4 $$det[a_{jk}]=\sum _{\mathrm{\ell }=1}^n{a}_{j\mathrm{\ell }}{A}_{j\mathrm{\ell }}.$$

ⓘ

Symbols:

$det$: determinant, $j$: integer, $k$: integer, $n$: nonnegative integer and $A_{jk}$: cofactor

Permalink:

http://dlmf.nist.gov/1.3.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §1.3(i), §1.3(i), §1.3 and Ch.1

… ►The determinant of an upper or lower triangular, or diagonal, square matrix $𝐀$ is the product of the diagonal elements $det(𝐀)={\prod }_{i=1}^n{a}_{ii}$. … ►

1.3.9 $$det{[a_{jk}]}^2\le \left(\sum _{k=1}^n{a}_{1k}^2\right)\left(\sum _{k=1}^n{a}_{2k}^2\right)\mathrm{\dots }\left(\sum _{k=1}^n{a}_{nk}^2\right).$$

ⓘ

Symbols:

$det$: determinant, $j$: integer, $k$: integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/1.3.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §1.3(i), §1.3(i), §1.3 and Ch.1

… ►for every distinct pair of $j,k$, or when one of the factors ${\sum }_{k=1}^n{a}_{jk}^2$ vanishes. … ►Square matices can be seen as linear operators because $𝐀(\alpha 𝐚+\beta 𝐛)=\alpha 𝐀𝐚+\beta 𝐀𝐛$ for all $\alpha ,\beta \in ℂ$ and $𝐚,𝐛\in {𝐄}_n$, the space of all $n$-dimensional vectors. …

32: 1.4 Calculus of One Variable

… ►where the sum is over all nonnegative integers $m_1,m_2,\mathrm{\dots },m_n$ that satisfy $m_1+2m_2+\mathrm{\cdots }+n{m}_n=n$, and $k=m_1+m_2+\mathrm{\cdots }+m_n$. … ►Definite integrals over the Stieltjes measure $d\alpha (x)$ could represent a sum, an integral, or a combination of the two. Let $d\alpha (x)=w(x)dx+{\sum }_{n=1}^N{w}_n\delta \left(x-x_n\right)dx$, $x_n\in (a,b)$, $n=1,\mathrm{\dots }N$. … ►

Square-Integrable Functions

►A function $f(x)$ is square-integrable if …

33: 36.12 Uniform Approximation of Integrals

… ►

36.12.3 $$I(𝐲,k)=\frac{\exp \left(\mathrm{i}kA(𝐲)\right)}{k^{1/(K+2)}}\sum _{m=0}^K\frac{a_m(𝐲)}{k^{m/(K+2)}}\left({\delta }_{m,0}-\left(1-{\delta }_{m,0}\right)\mathrm{i}\frac{\partial }{\partial z_m}\right){\mathrm{\Psi }}_K\left(𝐳(𝐲;k)\right)\left(1+O\left(\frac{1}{k}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, ${\delta }_{j,k}$: Kronecker delta, ${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: integer, $k$: variable, $K$: codimension, $I(𝐲,k)$: oscillatory integral, $a_m(𝐲)$: function, $A(𝐲)$: function, $𝐳(𝐲,k)$: function and $z_m(𝐲,k)$: function

Permalink:

http://dlmf.nist.gov/36.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §36.12(i), §36.12 and Ch.36

… ►

36.12.8 $$a_m(𝐲)=\sum _{n=1}^{K+1}\frac{P_{mn}(𝐲)G_n(𝐲)}{{(t_n(𝐱(𝐲)))}^{m+1}\prod _{\begin{array}{c}l=1\\ l\ne n\end{array}}^{K+1}(t_n(𝐱(𝐲))-t_l(𝐱(𝐲)))},$$

ⓘ

Defines:

$a_m(𝐲)$: function (locally)

Symbols:

$l$: integer, $n$: integer, $m$: integer, $K$: codimension, $𝐱(𝐲)$: function and $t_j(𝐱)$: solutions

Permalink:

http://dlmf.nist.gov/36.12.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §36.12(i), §36.12 and Ch.36

… ►

36.12.9 $$P_{mn}(𝐲)={(t_n(𝐱(𝐲)))}^{K+1}+\sum _{l=m+2}^K\frac{l}{K+2}{x}_l(𝐲){(t_n(𝐱(𝐲)))}^{l-1},$$

ⓘ

Symbols:

$l$: integer, $n$: integer, $m$: integer, $K$: codimension, $x_{\mathrm{i}}$: real parameter, $𝐱(𝐲)$: function and $t_j(𝐱)$: solutions

Permalink:

http://dlmf.nist.gov/36.12.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §36.12(i), §36.12 and Ch.36

… ►The square roots are real and positive when $𝐲$ is such that all the critical points are real, and are defined by analytic continuation elsewhere. …

34: 1.17 Integral and Series Representations of the Dirac Delta

… ►In the language of physics and applied mathematics, these equations indicate the normalizations chosen for these non-$L^2$ improper eigenfunctions of the differential operators (with derivatives respect to spatial co-ordinates) which generate them; the normalizations (1.17.12_1) and (1.17.12_2) are explicitly derived in Friedman (1990, Ch. 4), the others follow similarly. … ►The sum ${\sum }_{k=-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{\mathrm{i}k(x-a)}$ does not converge, but (1.17.18) can be interpreted as a generalized integral in the sense that … ►

1.17.21 $$\delta \left(x-a\right)=\frac{1}{2\pi }\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{\mathrm{i}k(x-a)}.$$

ⓘ

Symbols:

$\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $k$: integer

Referenced by:

§1.18(v)

Permalink:

http://dlmf.nist.gov/1.17.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §1.17(iii), §1.17 and Ch.1

… ►

1.17.22 $$\delta \left(x-a\right)=\sum _{k=0}^{\mathrm{\infty }}(k+\frac{1}{2})P_k\left(x\right)P_k\left(a\right).$$

ⓘ

Symbols:

$\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $P_n\left(x\right)$: Legendre polynomial and $k$: integer

Referenced by:

§1.17(iii), §1.17(iv), §14.18(iii)

Permalink:

http://dlmf.nist.gov/1.17.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §1.17(iii), §1.17(iii), §1.17 and Ch.1

… ►

1.17.25 $$\delta \left(\cos {\theta }_1-\cos {\theta }_2\right)\delta \left({\varphi }_1-{\varphi }_2\right)=\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}\sum _{m=-\mathrm{\ell }}^{\mathrm{\ell }}{Y}_{\mathrm{\ell },m}({\theta }_1,{\varphi }_1)\overline{Y_{\mathrm{\ell },m}({\theta }_2,{\varphi }_2)}.$$

ⓘ

Symbols:

$\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $\overline{z}$: complex conjugate, $\cos z$: cosine function, $Y_{l,m}(\theta ,\varphi )$: spherical harmonic and $m$: nonnegative integer

Referenced by:

§1.17(iii)

Permalink:

http://dlmf.nist.gov/1.17.E25

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.19):

As a notational clarification, the second spherical harmonic in the sum over $\mathrm{\ell }$ has been explicity marked up as a complex conjugate.

See also:

Annotations for §1.17(iii), §1.17(iii), §1.17 and Ch.1

…

35: 35.7 Gaussian Hypergeometric Function of Matrix Argument

… ►

35.7.3

ⓘ

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, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $a$: complex variable, $t_j$: eigenvalues of $𝐓$ , $b$: complex variable, $c$: complex variable and $k$: nonnegative integer

Referenced by:

Erratum (V1.0.25) for Equation (35.7.3)

Permalink:

http://dlmf.nist.gov/35.7.E3

Encodings:

pMML, png, TeX

Notational change (effective with 1.0.25):

Originally the matrix in the argument of the Gaussian hypergeometric function of matrix argument ${}_2{}^{}F_1^{}$ was written with round brackets. This matrix has been rewritten with square brackets to be consistent with the rest of the DLMF.

See also:

Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35

…

36: 10.25 Definitions

… ►

10.25.2 $$I_{\nu }\left(z\right)={(\frac{1}{2}z)}^{\nu }\sum _{k=0}^{\mathrm{\infty }}\frac{{(\frac{1}{4}{z}^2)}^k}{k!\mathrm{\Gamma }\left(\nu +k+1\right)}.$$

ⓘ

Defines:

$I_{\nu }\left(z\right)$: modified Bessel function of the first kind

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $!$: factorial (as in $n!$), $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.10

Referenced by:

§10.30(i), §10.31, §10.38, §10.45, §10.46, §10.53, §8.6(i)

Permalink:

http://dlmf.nist.gov/10.25.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.25(ii), §10.25 and Ch.10

… ►The principal branch corresponds to the principal value of the square root in (10.25.3), is analytic in $ℂ\setminus (-\mathrm{\infty },0]$, and two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi$. …

37: 19.18 Derivatives and Differential Equations

… ►

$w_j=b_j/{\displaystyle \sum _{j=1}^n}{b}_j,$

►

$a^{\prime }=-a+{\displaystyle \sum _{j=1}^n}{b}_j.$

… ►

19.18.6 $$\left(\frac{\partial }{\partial x}+\frac{\partial }{\partial y}+\frac{\partial }{\partial z}\right)R_F(x,y,z)=\frac{-1}{2\sqrt{x}\sqrt{y}\sqrt{z}},$$

ⓘ

Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ and $\partial x$: partial differential of $x$

Referenced by:

§19.18(ii), §19.32, Erratum (V1.1.3) for Chapter 19

Permalink:

http://dlmf.nist.gov/19.18.E6

Encodings:

pMML, png, TeX

Correction (effective with 1.1.3):

The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.

Suggested 2021-06-07 by Luc Maisonobe

See also:

Annotations for §19.18(ii), §19.18 and Ch.19

… ►

19.18.8 $$\sum _{j=1}^n{\partial }_j{R}_{-a}(𝐛;𝐳)=-a{R}_{-a-1}(𝐛;𝐳).$$

ⓘ

Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, $\partial x$: partial differential of $x$ and $n$: nonnegative integer

Referenced by:

§19.18(i), §19.18(ii)

Permalink:

http://dlmf.nist.gov/19.18.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.18(ii), §19.18 and Ch.19

… ►

19.18.11 $$\sum _{j=1}^n{z}_j{\partial }_{j}v=-av,$$

ⓘ

Symbols:

$\partial x$: partial differential of $x$ , $n$: nonnegative integer and $v$

Referenced by:

§19.18(ii), §19.18(ii)

Permalink:

http://dlmf.nist.gov/19.18.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §19.18(ii), §19.18 and Ch.19

…

38: 21.7 Riemann Surfaces

… ►

21.7.8 $$\zeta ={\sum _{j=1}^g{\omega }_j\frac{\partial }{\partial z_j}\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)|}_{𝐳=\mathrm{𝟎}}.$$

ⓘ

Symbols:

$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $g$: positive integer, $𝛀$: a Riemann matrix, $𝜶$: $g$-dimensional vector, $𝜷$: $g$-dimensional vector and $\omega$: differential

Permalink:

http://dlmf.nist.gov/21.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §21.7(ii), §21.7 and Ch.21

… ►Either branch of the square roots may be chosen, as long as the branch is consistent across $\mathrm{\Gamma }$. … ►

21.7.17 $$\sum _{P_j\in U}\prod _{k=1}^4\theta \left[\genfrac{}{}{0.0pt}{}{{𝜶}_k+{𝜼}^1(P_j)}{{𝜷}_k+{𝜼}^2(P_j)}\right]\left({𝐳}_k\right|𝛀)=\sum _{P_j\in U^c}\prod _{k=1}^4\theta \left[\genfrac{}{}{0.0pt}{}{{𝜶}_k+{𝜼}^1(P_j)}{{𝜷}_k+{𝜼}^2(P_j)}\right]\left({𝐳}_k\right|𝛀).$$

ⓘ

Symbols:

$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics, $\in$: element of, $𝛀$: a Riemann matrix, $𝜶$: $g$-dimensional vector, $𝜷$: $g$-dimensional vector, $P_j$: branch points and $U$: subset of $B$

Permalink:

http://dlmf.nist.gov/21.7.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §21.7(iii), §21.7 and Ch.21

39: 19.21 Connection Formulas

… ►

19.21.7 $$(x-y)R_D(y,z,x)+(z-y)R_D(x,y,z)=3R_F(x,y,z)-3y^{1/2}{x}^{-1/2}{z}^{-1/2},$$

ⓘ

Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables and $R_F(x,y,z)$: symmetric elliptic integral of first kind

Referenced by:

§19.21(ii), §19.25(i), Erratum (V1.1.3) for Chapter 19

Permalink:

http://dlmf.nist.gov/19.21.E7

Encodings:

pMML, png, TeX

Correction (effective with 1.1.3):

The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.

Suggested 2021-06-07 by Luc Maisonobe

See also:

Annotations for §19.21(ii), §19.21 and Ch.19

… ►

19.21.8 $$R_D(y,z,x)+R_D(z,x,y)+R_D(x,y,z)=3x^{-1/2}{y}^{-1/2}{z}^{-1/2},$$

ⓘ

Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables

Referenced by:

§19.21(ii), §19.33(iii), Erratum (V1.1.3) for Chapter 19

Permalink:

http://dlmf.nist.gov/19.21.E8

Encodings:

pMML, png, TeX

Correction (effective with 1.1.3):

The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.

Suggested 2021-06-07 by Luc Maisonobe

See also:

Annotations for §19.21(ii), §19.21 and Ch.19

… ►

19.21.10 $$2R_G(x,y,z)=z{R}_F(x,y,z)-\frac{1}{3}(x-z)(y-z)R_D(x,y,z)+x^{1/2}{y}^{1/2}{z}^{-1/2},$$ $z\ne 0$.

ⓘ

Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind and $R_G(x,y,z)$: symmetric elliptic integral of second kind

Referenced by:

§19.20(ii), §19.21(i), §19.21(ii), §19.21(ii), §19.36(i), §19.36(ii), Erratum (V1.1.3) for Chapter 19, Erratum (V1.1.7) for Equation (19.21.10)

Permalink:

http://dlmf.nist.gov/19.21.E10

Encodings:

pMML, png, TeX

Correction (effective with 1.1.7):

An incorrect factor of 3 was removed in the last term on the right-hand side.

Suggested 2022-06-27 by Abdulhafeez Abdulsalam

Correction (effective with 1.1.3):

The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.

Suggested 2021-06-07 by Luc Maisonobe

See also:

Annotations for §19.21(ii), §19.21 and Ch.19

… ►

19.21.11 $$6R_G(x,y,z)=3(x+y+z)R_F(x,y,z)-\sum x^2{R}_D(y,z,x)=\sum x(y+z)R_D(y,z,x),$$

ⓘ

Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind and $R_G(x,y,z)$: symmetric elliptic integral of second kind

Referenced by:

§19.21(i), §19.21(ii)

Permalink:

http://dlmf.nist.gov/19.21.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §19.21(ii), §19.21 and Ch.19

…

40: 22.16 Related Functions

… ►

22.16.9 $$\mathrm{am}(x,k)=\frac{\pi }{2K}x+2\sum _{n=1}^{\mathrm{\infty }}\frac{q^n\sin \left(2n\zeta \right)}{n(1+q^{2n})}.$$

ⓘ

Symbols:

$\mathrm{am}(x,k)$: Jacobi’s amplitude function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $q$: nome, $\sin z$: sine function, $x$: real and $k$: modulus

Referenced by:

§22.20(vi)

Permalink:

http://dlmf.nist.gov/22.16.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §22.16(i), §22.16(i), §22.16 and Ch.22

… ►See Figure 22.16.2. … ►

22.16.26 $$ℰ(x,k)=-{\int }_0^x\left({\mathrm{cs}}^2(t,k)-t^{-2}\right)dt+x^{-1}-\mathrm{cn}(x,k)\mathrm{ds}(x,k).$$

ⓘ

Symbols:

$ℰ(x,k)$: Jacobi’s epsilon function, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{cs}(z,k)$: Jacobian elliptic function, $\mathrm{ds}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $x$: real and $k$: modulus

A&S Ref:

16.26.12

Referenced by:

§22.16(ii)

Permalink:

http://dlmf.nist.gov/22.16.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §22.16(ii), §22.16(ii), §22.16 and Ch.22

…