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21—30 of 136 matching pages

21: 10.34 Analytic Continuation

… ►For complex

\nu

replace

\nu

\overline{\nu}

on the right-hand sides.

22: 14.17 Integrals

… ►

14.17.6

\int_{-1}^{1}\mathsf{P}^{m}_{l}\left(x\right)\mathsf{P}^{m}_{n}\left(x\right)% \,\mathrm{d}x=\frac{(n+m)!}{(n-m)!\left(n+\frac{1}{2}\right)}\delta_{l,n},

ⓘ

Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $!$ : factorial (as in $n!$ ), $\int$ : integral, $x$ : real variable, $m$ : nonnegative integer, $n$ : nonnegative integer and $l$ : nonnegative integer
A&S Ref:: 8.14.11 8.14.13
Referenced by:: §14.17(iii), Erratum (V1.0.17) for Equations (10.22.37), (10.22.38), (14.17.6)–(14.17.9)
Permalink:: http://dlmf.nist.gov/14.17.E6
Encodings:: pMML, png, TeX
Clarification (effective with 1.0.17):: The Kronecker delta symbol has been moved furthest to the right.
See also:: Annotations for §14.17(iii), §14.17 and Ch.14

►

14.17.7

\int_{-1}^{1}\mathsf{P}^{m}_{l}\left(x\right)\mathsf{P}^{-m}_{n}\left(x\right)% \,\mathrm{d}x=\frac{(-1)^{m}}{l+\frac{1}{2}}\delta_{l,n},

ⓘ

Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : real variable, $m$ : nonnegative integer, $n$ : nonnegative integer and $l$ : nonnegative integer
Referenced by:: §14.17(iii)
Permalink:: http://dlmf.nist.gov/14.17.E7
Encodings:: pMML, png, TeX
Clarification (effective with 1.0.17):: The Kronecker delta symbol has been moved furthest to the right and $(-1)^{m}$ has been moved to the numerator of the fraction.
See also:: Annotations for §14.17(iii), §14.17 and Ch.14

►

14.17.8

\int_{-1}^{1}\frac{\mathsf{P}^{l}_{n}\left(x\right)\mathsf{P}^{m}_{n}\left(x% \right)}{1-x^{2}}\,\mathrm{d}x=\frac{(n+m)!}{(n-m)!m}\delta_{l,m},

m>0

ⓘ

Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $!$ : factorial (as in $n!$ ), $\int$ : integral, $x$ : real variable, $m$ : nonnegative integer, $n$ : nonnegative integer and $l$ : nonnegative integer
A&S Ref:: 8.14.12 8.14.14
Referenced by:: §14.17(iii)
Permalink:: http://dlmf.nist.gov/14.17.E8
Encodings:: pMML, png, TeX
Clarification (effective with 1.0.17):: The Kronecker delta symbol has been moved furthest to the right.
See also:: Annotations for §14.17(iii), §14.17 and Ch.14

►

14.17.9

\int_{-1}^{1}\frac{\mathsf{P}^{l}_{n}\left(x\right)\mathsf{P}^{-m}_{n}\left(x% \right)}{1-x^{2}}\,\mathrm{d}x=\frac{(-1)^{l}}{l}\delta_{l,m},

l>0

ⓘ

Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $x$ : real variable, $m$ : nonnegative integer, $n$ : nonnegative integer and $l$ : nonnegative integer
Referenced by:: §14.17(iii), Erratum (V1.0.17) for Equations (10.22.37), (10.22.38), (14.17.6)–(14.17.9)
Permalink:: http://dlmf.nist.gov/14.17.E9
Encodings:: pMML, png, TeX
Clarification (effective with 1.0.17):: The Kronecker delta symbol has been moved furthest to the right and $(-1)^{l}$ has been moved to the numerator of the fraction.
See also:: Annotations for §14.17(iii), §14.17 and Ch.14

…

23: 18.7 Interrelations and Limit Relations

… ►

18.7.5

V_{n}\left(x\right)=\ifrac{P^{(-\frac{1}{2},\frac{1}{2})}_{n}\left(x\right)}{P% ^{(-\frac{1}{2},\frac{1}{2})}_{n}\left(1\right)},

ⓘ

Symbols:: $W_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the fourth kind, $V_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the third kind, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.17(viii), §18.5(i), (18.7.5), (18.7.6), §18.7(i), §18.7(ii), Erratum (V1.0.28) for Table 18.3.1
Permalink:: http://dlmf.nist.gov/18.7.E5
Encodings:: pMML, png, TeX
Correction (effective with 1.0.28):: The DLMF now adopts the definitions for the Chebyshev polynomials of the third and fourth kinds $V_{n}\left(x\right)$ , $W_{n}\left(x\right)$ used in Mason and Handscomb (2003). Therefore $V_{n}\left(x\right)$ , $W_{n}\left(x\right)$ , having been interchanged, the right-hand sides of (18.7.5) and (18.7.6) have been interchanged. For further details see Errata.
See also:: Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

►

18.7.6

W_{n}\left(x\right)=\ifrac{(2n+1)P^{(\frac{1}{2},-\frac{1}{2})}_{n}\left(x% \right)}{P^{(\frac{1}{2},-\frac{1}{2})}_{n}\left(1\right)}.

ⓘ

Symbols:: $W_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the fourth kind, $V_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the third kind, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.18(i), §18.3, §18.5(iii), §18.5(i), (18.7.5), (18.7.6), §18.7(i), §18.7(ii), Erratum (V1.0.28) for Table 18.3.1
Permalink:: http://dlmf.nist.gov/18.7.E6
Encodings:: pMML, png, TeX
Correction (effective with 1.0.28):: The DLMF now adopts the definitions for the Chebyshev polynomials of the third and fourth kinds $V_{n}\left(x\right)$ , $W_{n}\left(x\right)$ used in Mason and Handscomb (2003). Therefore $V_{n}\left(x\right)$ , $W_{n}\left(x\right)$ , having been interchanged, the right-hand sides of (18.7.5) and (18.7.6) have been interchanged. For further details see Errata.
See also:: Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

… ►

18.7.17

U_{2n}\left(x\right)=W_{n}\left(2x^{2}-1\right),

ⓘ

Symbols:: $W_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the fourth kind, $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, $V_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the third kind, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.5.30
Referenced by:: §18.7(ii), Erratum (V1.0.28) for Table 18.3.1
Permalink:: http://dlmf.nist.gov/18.7.E17
Encodings:: pMML, png, TeX
Correction (effective with 1.0.28):: The DLMF now adopts the definitions for the Chebyshev polynomials of the third and fourth kinds $V_{n}\left(x\right)$ , $W_{n}\left(x\right)$ used in Mason and Handscomb (2003). Therefore $V_{n}\left(x\right)$ , $W_{n}\left(x\right)$ , having been interchanged, on the right-hand side we replaced $V_{n}\left(2x^{2}-1\right)$ , with $W_{n}\left(2x^{2}-1\right)$ . For further details see Errata.
See also:: Annotations for §18.7(ii), §18.7 and Ch.18

►

18.7.18

T_{2n+1}\left(x\right)=xV_{n}\left(2x^{2}-1\right).

ⓘ

Symbols:: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $W_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the fourth kind, $V_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the third kind, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.5.29
Referenced by:: §18.7(ii), Erratum (V1.0.28) for Table 18.3.1
Permalink:: http://dlmf.nist.gov/18.7.E18
Encodings:: pMML, png, TeX
Correction (effective with 1.0.28):: The DLMF now adopts the definitions for the Chebyshev polynomials of the third and fourth kinds $V_{n}\left(x\right)$ , $W_{n}\left(x\right)$ used in Mason and Handscomb (2003). Therefore $V_{n}\left(x\right)$ , $W_{n}\left(x\right)$ , having been interchanged, on the right-hand side we replaced $xW_{n}\left(2x^{2}-1\right)$ , with $xV_{n}\left(2x^{2}-1\right)$ . For further details see Errata.
See also:: Annotations for §18.7(ii), §18.7 and Ch.18

…

24: 21.3 Symmetry and Quasi-Periodicity

… ►

21.3.4

\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}+\mathbf{m}_{1}}{% \boldsymbol{{\beta}}+\mathbf{m}_{2}}\left(\mathbf{z}\middle|\boldsymbol{{% \Omega}}\right)=e^{2\pi i\boldsymbol{{\alpha}}\cdot\mathbf{m}_{2}}\theta% \genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}\left(% \mathbf{z}\middle|\boldsymbol{{\Omega}}\right).

ⓘ

Symbols:: $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function with characteristics, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : $g$ -dimensional vector and $\boldsymbol{{\beta}}$ : $g$ -dimensional vector
Referenced by:: Erratum (V1.0.5) for Equation (21.3.4)
Permalink:: http://dlmf.nist.gov/21.3.E4
Encodings:: pMML, png, TeX
Errata (effective with 1.0.5):: Originally the vector $\mathbf{m}_{2}$ on the right-hand-side was given incorrectly as $\mathbf{m}_{1}$ .
Reported 2012-08-27 by Klaas Vantournhout
See also:: Annotations for §21.3(ii), §21.3 and Ch.21

…

25: 26.13 Permutations: Cycle Notation

… ►A permutation that consists of a single cycle of length

k

can be written as the composition of

k-1

two-cycles (read from right to left): …

26: 27.4 Euler Products and Dirichlet Series

… ►In this case the infinite product on the right (extended over all primes

p

) is also absolutely convergent and is called the Euler product of the series. …

27: 28.26 Asymptotic Approximations for Large $q$

… ►The asymptotic expansions of

\mathrm{Fs}_{m}\left(z,h\right)

and

\mathrm{Gs}_{m}\left(z,h\right)

in the same circumstances are also given by the right-hand sides of (28.26.4) and (28.26.5), respectively. …

28: 31.8 Solutions via Quadratures

…

29: 33.6 Power-Series Expansions in $\rho$

… ►

33.6.5

{H^{\pm}_{\ell}}\left(\eta,\rho\right)=\frac{e^{\pm\mathrm{i}{\theta_{\ell}}% \left(\eta,\rho\right)}}{(2\ell+1)!\Gamma\left(-\ell\pm\mathrm{i}\eta\right)}% \left(\sum_{k=0}^{\infty}\frac{{\left(a\right)_{k}}}{{\left(2\ell+2\right)_{k}% }k!}(\mp 2\mathrm{i}\rho)^{a+k}\left(\ln\left(\mp 2\mathrm{i}\rho\right)+\psi% \left(a+k\right)-\psi\left(1+k\right)-\psi\left(2\ell+2+k\right)\right)-\sum_{% k=1}^{2\ell+1}\frac{(2\ell+1)!(k-1)!}{(2\ell+1-k)!{\left(1-a\right)_{k}}}(\mp 2% \mathrm{i}\rho)^{a-k}\right),

ⓘ

Symbols:: ${\theta_{\NVar{\ell}}}\left(\NVar{\eta},\NVar{\rho}\right)$ : phase of Coulomb functions, $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, ${H^{\NVar{\pm}}_{\NVar{\ell}}}\left(\NVar{\eta},\NVar{\rho}\right)$ : irregular Coulomb radial functions, $\ln\NVar{z}$ : principal branch of logarithm function, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable, $\eta$ : real parameter and $a$ : variable
A&S Ref:: 14.1.14 (equivalent)
Referenced by:: §33.13, §33.6, §33.6, Erratum (V1.0.19) for Equation (33.6.5)
Permalink:: http://dlmf.nist.gov/33.6.E5
Encodings:: pMML, png, TeX
Errata (effective with 1.0.19):: On the right-hand side the factor in the denominator was incorrectly written as $\Gamma\left(-\ell+\mathrm{i}\eta\right)$ . This has been corrected to $\Gamma\left(-\ell\pm\mathrm{i}\eta\right)$ .
Reported 2018-05-15 by Ian Thompson
See also:: Annotations for §33.6 and Ch.33

…

30: 11.9 Lommel Functions

… ►the right-hand side being replaced by its limiting form when

\mu\pm\nu

is an odd negative integer. … ►If either of

\mu\pm\nu

equals an odd positive integer, then the right-hand side of (11.9.9) terminates and represents

S_{{\mu},{\nu}}\left(z\right)

exactly. …