§18.17 Integrals

18.17.1		$$\begin{array}{l}2n{\int }_0^x{(1-y)}^{\alpha }{(1+y)}^{\beta }{P}_n^{(\alpha ,\beta )}\left(y\right)dy\\ =P_{n-1}^{(\alpha +1,\beta +1)}\left(0\right)-{(1-x)}^{\alpha +1}{(1+x)}^{\beta +1}{P}_{n-1}^{(\alpha +1,\beta +1)}\left(x\right).\end{array}$$
ⓘ Symbols: $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $dx$: differential of $x$, $\int$: integral, $y$: real variable, $n$: nonnegative integer and $x$: real variable Referenced by: §18.17(i), §18.17(i) Permalink: http://dlmf.nist.gov/18.17.E1 Encodings: TeX, pMML, png See also: Annotations for §18.17(i), §18.17(i), §18.17 and Ch.18

18.17.2		$${\int }_0^x{L}_m\left(y\right)L_n\left(x-y\right)dy={\int }_0^x{L}_{m+n}\left(y\right)dy=L_{m+n}\left(x\right)-L_{m+n+1}\left(x\right).$$
ⓘ Symbols: $dx$: differential of $x$, $\int$: integral, $L_n\left(x\right)=L_n^{(0)}\left(x\right)$: Laguerre polynomial, $y$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Referenced by: §18.17(i) Permalink: http://dlmf.nist.gov/18.17.E2 Encodings: TeX, pMML, png See also: Annotations for §18.17(i), §18.17(i), §18.17 and Ch.18

18.17.3		$${\int }_0^x{H}_n\left(y\right)dy=\frac{1}{2(n+1)}(H_{n+1}\left(x\right)-H_{n+1}\left(0\right)),$$
ⓘ Symbols: $H_n\left(x\right)$: Hermite polynomial, $dx$: differential of $x$, $\int$: integral, $y$: real variable, $n$: nonnegative integer and $x$: real variable Referenced by: §18.17(i), §18.17(i) Permalink: http://dlmf.nist.gov/18.17.E3 Encodings: TeX, pMML, png See also: Annotations for §18.17(i), §18.17(i), §18.17 and Ch.18

18.17.4		$${\int }_0^x{\mathrm{e}}^{-y^2}{H}_n\left(y\right)dy=H_{n-1}\left(0\right)-{\mathrm{e}}^{-x^2}{H}_{n-1}\left(x\right).$$
ⓘ Symbols: $H_n\left(x\right)$: Hermite polynomial, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $y$: real variable, $n$: nonnegative integer and $x$: real variable Referenced by: §18.17(i), §18.17(i) Permalink: http://dlmf.nist.gov/18.17.E4 Encodings: TeX, pMML, png See also: Annotations for §18.17(i), §18.17(i), §18.17 and Ch.18

Just as the indefinite integrals (18.17.1), (18.17.3) and (18.17.4), many similar formulas can be obtained by applying (1.4.26) to the differentiation formulas (18.9.15), (18.9.16) and (18.9.19)–(18.9.28).

18.17.5		$$\begin{array}{l}\frac{C_n^{(\lambda )}\left(\cos {\theta }_1\right)}{C_n^{(\lambda )}\left(1\right)}\frac{C_n^{(\lambda )}\left(\cos {\theta }_2\right)}{C_n^{(\lambda )}\left(1\right)}\\ =\frac{\mathrm{\Gamma }\left(\lambda +\frac{1}{2}\right)}{{\pi }^{\frac{1}{2}}\mathrm{\Gamma }\left(\lambda \right)}{\int }_0^{\pi }\frac{C_n^{(\lambda )}\left(\cos {\theta }_1\cos {\theta }_2+\sin {\theta }_1\sin {\theta }_2\cos \varphi \right)}{C_n^{(\lambda )}\left(1\right)}{(\sin \varphi )}^{2\lambda -1}d\varphi ,\end{array}$$
$\lambda >0$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\int$: integral, $\sin z$: sine function, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial and $n$: nonnegative integer Referenced by: §18.17(ii), §18.17(ii), §18.18(ii) Permalink: http://dlmf.nist.gov/18.17.E5 Encodings: TeX, pMML, png See also: Annotations for §18.17(ii), §18.17(ii), §18.17 and Ch.18

18.17.6		$$P_n\left(\cos {\theta }_1\right)P_n\left(\cos {\theta }_2\right)=\frac{1}{\pi }{\int }_0^{\pi }{P}_n\left(\cos {\theta }_1\cos {\theta }_2+\sin {\theta }_1\sin {\theta }_2\cos \varphi \right)d\varphi .$$
ⓘ Symbols: $P_n\left(x\right)$: Legendre polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\int$: integral, $\sin z$: sine function and $n$: nonnegative integer Referenced by: §18.17(ii), §18.17(ii), §18.18(ii) Permalink: http://dlmf.nist.gov/18.17.E6 Encodings: TeX, pMML, png See also: Annotations for §18.17(ii), §18.17(ii), §18.17 and Ch.18

For formulas for Jacobi and Laguerre polynomials analogous to (18.17.5) and (18.17.6), see Koornwinder (1974, 1977). For addition formulas corresponding to (18.17.5) and (18.17.6) see (18.18.8) and (18.18.9), respectively.

18.17.7		$${\left(P_n\left(x\right)\right)}^2+4{\pi }^{-2}{\left({𝖰}_n\left(x\right)\right)}^2=4{\pi }^{-2}{\int }_1^{\mathrm{\infty }}{Q}_n\left(x^2+(1-x^2)t\right){(t^2-1)}^{-\frac{1}{2}}dt,$$
.
ⓘ Symbols: $P_n\left(x\right)$: Legendre polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\int$: integral, ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: Legendre function of the second kind, $t$: real variable, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.17.E7 Encodings: TeX, pMML, png See also: Annotations for §18.17(iii), §18.17(iii), §18.17 and Ch.18

For the Ferrers function ${𝖰}_n\left(x\right)$ and Legendre function $Q_n\left(x\right)$ see §§14.3(i) and 14.3(ii), with $\mu =0$ and $\nu =n$.

18.17.8		$$\begin{array}{l}{\left(H_n\left(x\right)\right)}^2+2^n{(n!)}^2{\mathrm{e}}^{x^2}{\left(V(-n-\frac{1}{2},2^{\frac{1}{2}}x)\right)}^2\\ =\frac{2^{n+\frac{3}{2}}n!{\mathrm{e}}^{x^2}}{\pi }{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-(2n+1)t+x^2\tanh t}}{{(\sinh 2t)}^{\frac{1}{2}}}dt.\end{array}$$
ⓘ Symbols: $H_n\left(x\right)$: Hermite polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\sinh z$: hyperbolic sine function, $\tanh z$: hyperbolic tangent function, $\int$: integral, $V(a,z)$: parabolic cylinder function, $t$: real variable, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.17.E8 Encodings: TeX, pMML, png See also: Annotations for §18.17(iii), §18.17(iii), §18.17 and Ch.18

For the parabolic cylinder function $V(a,z)$ see §12.2. For similar formulas for ultraspherical polynomials see Durand (1975), and for Jacobi and Laguerre polynomials see Durand (1978).

18.17.9		$$\frac{{(1-x)}^{\alpha +\mu }{P}_n^{(\alpha +\mu ,\beta -\mu )}\left(x\right)}{\mathrm{\Gamma }\left(\alpha +\mu +n+1\right)}={\int }_x^1\frac{{(1-y)}^{\alpha }{P}_n^{(\alpha ,\beta )}\left(y\right)}{\mathrm{\Gamma }\left(\alpha +n+1\right)}\frac{{(y-x)}^{\mu -1}}{\mathrm{\Gamma }\left(\mu \right)}dy,$$
$\mu >0$, ,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $dx$: differential of $x$, $\int$: integral, $y$: real variable, $n$: nonnegative integer and $x$: real variable Referenced by: §1.15(vi), §18.17(iv), §18.17(iv), §18.17(viii) Permalink: http://dlmf.nist.gov/18.17.E9 Encodings: TeX, pMML, png See also: Annotations for §18.17(iv), §18.17(iv), §18.17 and Ch.18

18.17.10		${\displaystyle \frac{x^{\beta +\mu }{(x+1)}^n}{\mathrm{\Gamma }\left(\beta +\mu +n+1\right)}}{P}_n^{(\alpha ,\beta +\mu )}\left({\displaystyle \frac{x-1}{x+1}}\right)$	$={\displaystyle {\int }_0^x}{\displaystyle \frac{y^{\beta }{(y+1)}^n}{\mathrm{\Gamma }\left(\beta +n+1\right)}}{P}_n^{(\alpha ,\beta )}\left({\displaystyle \frac{y-1}{y+1}}\right){\displaystyle \frac{{(x-y)}^{\mu -1}}{\mathrm{\Gamma }\left(\mu \right)}}dy,$
$\mu >0$, $x>0$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $dx$: differential of $x$, $\int$: integral, $y$: real variable, $n$: nonnegative integer and $x$: real variable Referenced by: §18.17(iv), §18.17(iv) Permalink: http://dlmf.nist.gov/18.17.E10 Encodings: TeX, pMML, png See also: Annotations for §18.17(iv), §18.17(iv), §18.17 and Ch.18
18.17.11		${\displaystyle \frac{\mathrm{\Gamma }\left(n+\alpha +\beta -\mu +1\right)}{x^{n+\alpha +\beta -\mu +1}}}{P}_n^{(\alpha ,\beta -\mu )}\left(1-2x^{-1}\right)$	$={\displaystyle {\int }_x^{\mathrm{\infty }}}{\displaystyle \frac{\mathrm{\Gamma }\left(n+\alpha +\beta +1\right)}{y^{n+\alpha +\beta +1}}}{P}_n^{(\alpha ,\beta )}\left(1-2y^{-1}\right){\displaystyle \frac{{(y-x)}^{\mu -1}}{\mathrm{\Gamma }\left(\mu \right)}}dy,$
$\alpha +\beta +1>\mu >0$, $x>1$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $dx$: differential of $x$, $\int$: integral, $y$: real variable, $n$: nonnegative integer and $x$: real variable Referenced by: §1.15(vi), §18.17(iv), §18.17(iv) Permalink: http://dlmf.nist.gov/18.17.E11 Encodings: TeX, pMML, png See also: Annotations for §18.17(iv), §18.17(iv), §18.17 and Ch.18

and three formulas similar to (18.17.9)–(18.17.11) by symmetry; compare the second row in Table 18.6.1. Formula (18.17.9), after substitution of (18.5.7), is a special case of (15.6.8). Formulas (18.17.9), (18.17.10) and (18.17.11) are fractional generalizations of $n$-th derivative formulas which are, after substitution of (18.5.7), special cases of (15.5.4), (15.5.5) and (15.5.3), respectively.

18.17.12		${\displaystyle \frac{\mathrm{\Gamma }\left(\lambda -\mu \right)C_n^{(\lambda -\mu )}\left(x^{-\frac{1}{2}}\right)}{x^{\lambda -\mu +\frac{1}{2}n}}}$	$={\displaystyle {\int }_x^{\mathrm{\infty }}}{\displaystyle \frac{\mathrm{\Gamma }\left(\lambda \right)C_n^{(\lambda )}\left(y^{-\frac{1}{2}}\right)}{y^{\lambda +\frac{1}{2}n}}}{\displaystyle \frac{{(y-x)}^{\mu -1}}{\mathrm{\Gamma }\left(\mu \right)}}dy,$
$\lambda >\mu >0$, $x>0$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$, $\int$: integral, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $y$: real variable, $n$: nonnegative integer and $x$: real variable Referenced by: §18.17(iv) Permalink: http://dlmf.nist.gov/18.17.E12 Encodings: TeX, pMML, png See also: Annotations for §18.17(iv), §18.17(iv), §18.17 and Ch.18
18.17.13		${\displaystyle \frac{x^{\frac{1}{2}n}{(x-1)}^{\lambda +\mu -\frac{1}{2}}}{\mathrm{\Gamma }\left(\lambda +\mu +\frac{1}{2}\right)}}{\displaystyle \frac{C_n^{(\lambda +\mu )}\left(x^{-\frac{1}{2}}\right)}{C_n^{(\lambda +\mu )}\left(1\right)}}$	$={\displaystyle {\int }_1^x}{\displaystyle \frac{y^{\frac{1}{2}n}{(y-1)}^{\lambda -\frac{1}{2}}}{\mathrm{\Gamma }\left(\lambda +\frac{1}{2}\right)}}{\displaystyle \frac{C_n^{(\lambda )}\left(y^{-\frac{1}{2}}\right)}{C_n^{(\lambda )}\left(1\right)}}{\displaystyle \frac{{(x-y)}^{\mu -1}}{\mathrm{\Gamma }\left(\mu \right)}}dy,$
$\mu >0$, $x>1$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$, $\int$: integral, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $y$: real variable, $n$: nonnegative integer and $x$: real variable Referenced by: §1.15(vi), §18.17(iv) Permalink: http://dlmf.nist.gov/18.17.E13 Encodings: TeX, pMML, png See also: Annotations for §18.17(iv), §18.17(iv), §18.17 and Ch.18

Formulas (18.17.12) and (18.17.13) are fractional generalizations of the differentiation formulas given in Erdélyi et al. (1953b, §10.9(15)).

18.17.14		${\displaystyle \frac{x^{\alpha +\mu }{L}_n^{(\alpha +\mu )}\left(x\right)}{\mathrm{\Gamma }\left(\alpha +\mu +n+1\right)}}$	$={\displaystyle {\int }_0^x}{\displaystyle \frac{y^{\alpha }{L}_n^{(\alpha )}\left(y\right)}{\mathrm{\Gamma }\left(\alpha +n+1\right)}}{\displaystyle \frac{{(x-y)}^{\mu -1}}{\mathrm{\Gamma }\left(\mu \right)}}dy,$
$\mu >0$, $x>0$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $dx$: differential of $x$, $\int$: integral, $y$: real variable, $n$: nonnegative integer and $x$: real variable A&S Ref: 22.13.13 Referenced by: §18.17(iv) Permalink: http://dlmf.nist.gov/18.17.E14 Encodings: TeX, pMML, png See also: Annotations for §18.17(iv), §18.17(iv), §18.17 and Ch.18
18.17.15		${\mathrm{e}}^{-x}{L}_n^{(\alpha )}\left(x\right)$	$={\displaystyle {\int }_x^{\mathrm{\infty }}}{\mathrm{e}}^{-y}{L}_n^{(\alpha +\mu )}\left(y\right){\displaystyle \frac{{(y-x)}^{\mu -1}}{\mathrm{\Gamma }\left(\mu \right)}}dy,$
$\mu >0$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $y$: real variable, $n$: nonnegative integer and $x$: real variable Referenced by: §18.17(iv), §18.17(iv) Permalink: http://dlmf.nist.gov/18.17.E15 Encodings: TeX, pMML, png See also: Annotations for §18.17(iv), §18.17(iv), §18.17 and Ch.18

Formulas (18.17.14) and (18.17.15) are fractional generalizations of $n$-th derivative formulas which are, after substitution of (13.6.19), special cases of (13.3.18) and (13.3.20), respectively.

18.17.16		$$\begin{array}{l}{\int }_{-1}^1{(1-x)}^{\alpha }{(1+x)}^{\beta }{P}_n^{(\alpha ,\beta )}\left(x\right){\mathrm{e}}^{\mathrm{i}xy}dx\\ =\frac{{(\mathrm{i}y)}^n{\mathrm{e}}^{\mathrm{i}y}}{n!}{2}^{n+\alpha +\beta +1}\mathrm{B}(n+\alpha +1,n+\beta +1){}_1{}^{}F_1^{}(n+\alpha +1;2n+\alpha +\beta +2;-2\mathrm{i}y).\end{array}$$
ⓘ Symbols: $\mathrm{B}(a,b)$: beta function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${}_1{}^{}F_1^{}(a;b;z)$: $=M(a,b,z)$ notation for the Kummer confluent hypergeometric function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\int$: integral, $y$: real variable, $n$: nonnegative integer and $x$: real variable Referenced by: §18.17(v), §18.17(vi) Permalink: http://dlmf.nist.gov/18.17.E16 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

For the beta function $\mathrm{B}(a,b)$ see §5.12, and for the confluent hypergeometric function ${}_1{}^{}F_1^{}$ see (16.2.1) and Chapter 13.

18.17.16_5		$${\int }_{-1}^1{(1-x^2)}^{\lambda -\frac{1}{2}}{C}_n^{(\lambda )}\left(x\right){\mathrm{e}}^{\mathrm{i}xy}dx=\frac{2\pi {\mathrm{i}}^n\mathrm{\Gamma }\left(n+2\lambda \right)J_{n+\lambda }\left(y\right)}{n!\mathrm{\Gamma }\left(\lambda \right){(2y)}^{\lambda }},$$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: 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, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $y$: real variable, $n$: nonnegative integer and $x$: real variable Referenced by: §18.17(v), Erratum (V1.2.0) §18.17 Permalink: http://dlmf.nist.gov/18.17.E16_5 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

18.17.17		$${\int }_0^1{(1-x^2)}^{\lambda -\frac{1}{2}}{C}_{2n}^{(\lambda )}\left(x\right)\cos \left(xy\right)dx=\frac{{(-1)}^n\pi \mathrm{\Gamma }\left(2n+2\lambda \right)J_{\lambda +2n}\left(y\right)}{(2n)!\mathrm{\Gamma }\left(\lambda \right){(2y)}^{\lambda }},$$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $!$: factorial (as in $n!$), $\int$: integral, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $y$: real variable, $n$: nonnegative integer and $x$: real variable Referenced by: §18.17(v) Permalink: http://dlmf.nist.gov/18.17.E17 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

18.17.18		$${\int }_0^1{(1-x^2)}^{\lambda -\frac{1}{2}}{C}_{2n+1}^{(\lambda )}\left(x\right)\sin \left(xy\right)dx=\frac{{(-1)}^n\pi \mathrm{\Gamma }\left(2n+2\lambda +1\right)J_{2n+\lambda +1}\left(y\right)}{(2n+1)!\mathrm{\Gamma }\left(\lambda \right){(2y)}^{\lambda }}.$$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $!$: factorial (as in $n!$), $\int$: integral, $\sin z$: sine function, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $y$: real variable, $n$: nonnegative integer and $x$: real variable Referenced by: §18.17(v) Permalink: http://dlmf.nist.gov/18.17.E18 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

For the Bessel function $J_{\nu }$ see §10.2(ii).

18.17.19		$${\int }_{-1}^1{P}_n\left(x\right){\mathrm{e}}^{\mathrm{i}xy}dx={\mathrm{i}}^n\sqrt{\frac{2\pi }{y}}{J}_{n+\frac{1}{2}}\left(y\right),$$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $P_n\left(x\right)$: Legendre polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $y$: real variable, $n$: nonnegative integer and $x$: real variable Referenced by: §18.17(v) Permalink: http://dlmf.nist.gov/18.17.E19 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

18.17.20		$${\int }_0^1{P}_n\left(1-2x^2\right)\cos \left(xy\right)dx={(-1)}^n\frac{1}{2}\pi J_{n+\frac{1}{2}}\left(\frac{1}{2}y\right)J_{-n-\frac{1}{2}}\left(\frac{1}{2}y\right),$$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $P_n\left(x\right)$: Legendre polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\int$: integral, $y$: real variable, $n$: nonnegative integer and $x$: real variable Referenced by: §18.17(v) Permalink: http://dlmf.nist.gov/18.17.E20 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

18.17.21		$${\int }_0^1{P}_n\left(1-2x^2\right)\sin \left(xy\right)dx=\frac{1}{2}\pi {\left(J_{n+\frac{1}{2}}\left(\frac{1}{2}y\right)\right)}^2.$$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $P_n\left(x\right)$: Legendre polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\int$: integral, $\sin z$: sine function, $y$: real variable, $n$: nonnegative integer and $x$: real variable Referenced by: §18.17(v) Permalink: http://dlmf.nist.gov/18.17.E21 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

18.17.21_1		$$\frac{1}{\sqrt{2\pi c}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{1}{2}{x}^2/c}{H}_n\left(x\right){\mathrm{e}}^{\mathrm{i}xy}dx={\left(\mathrm{i}\sqrt{2c-1}\right)}^n{\mathrm{e}}^{-\frac{1}{2}c{y}^2}{H}_n\left(\frac{cy}{\sqrt{2c-1}}\right),$$
$\mathrm{\Re }\left(c\right)>0$, $c\ne \frac{1}{2}$,
ⓘ Symbols: $H_n\left(x\right)$: Hermite polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $y$: real variable, $n$: nonnegative integer and $x$: real variable Sources: Erdélyi et al. (1954a, §1.10(8), §2.10(10)); Erdélyi et al. (1953b, §10.13(30)) Proof sketch: In the integrand of the left-hand side substitute (18.10.8) with data for $H_n\left(x\right)$ given by Table 18.10.1. Interchange integrals. Then the inner integral can be evaluated. The resulting integral can again be evaluated by (18.10.8) for Hermite. Referenced by: §18.17(v), §18.17(v), Erratum (V1.2.0) §18.17 Permalink: http://dlmf.nist.gov/18.17.E21_1 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

18.17.21_2		$$\frac{1}{\sqrt{\pi }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-x^2}{H}_n\left(x\right){\mathrm{e}}^{\mathrm{i}xy}dx={\mathrm{i}}^n{\mathrm{e}}^{-\frac{1}{4}{y}^2}{y}^n,$$
ⓘ Symbols: $H_n\left(x\right)$: Hermite polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $y$: real variable, $n$: nonnegative integer and $x$: real variable Referenced by: §18.17(v) Permalink: http://dlmf.nist.gov/18.17.E21_2 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

18.17.21_3		$$\frac{1}{\sqrt{2\pi }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{1}{2}{x}^2}{H}_n\left(x\right){\mathrm{e}}^{\mathrm{i}xy}dx={\mathrm{i}}^n{\mathrm{e}}^{-\frac{1}{2}{y}^2}{H}_n\left(y\right).$$
ⓘ Symbols: $H_n\left(x\right)$: Hermite polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $y$: real variable, $n$: nonnegative integer and $x$: real variable Referenced by: §18.17(v), §18.17(v), Erratum (V1.2.0) §18.17 Permalink: http://dlmf.nist.gov/18.17.E21_3 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

In (18.17.21_1) the branch choice of $\sqrt{2c-1}$ for is unimportant because on the right-hand side only even powers of $\sqrt{2c-1}$ occur after expansion of the Hermite polynomial by (18.5.13). Formulas (18.17.21_2) and (18.17.21_3) are respectively the limit case $c\to \frac{1}{2}$ and the special case $c=1$ of (18.17.21_1).

18.17.22		$$\frac{1}{2\sqrt{\pi }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{1}{4}{x}^2}{\mathrm{𝐻𝑒}}_n\left(x\right){\mathrm{e}}^{\frac{1}{2}\mathrm{i}xy}dx={\mathrm{i}}^n{\mathrm{e}}^{-\frac{1}{4}{y}^2}{\mathrm{𝐻𝑒}}_n\left(y\right),$$
ⓘ Symbols: ${\mathrm{𝐻𝑒}}_n\left(x\right)$: Hermite polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $y$: real variable, $n$: nonnegative integer and $x$: real variable Referenced by: §18.17(v) Permalink: http://dlmf.nist.gov/18.17.E22 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

18.17.23		$${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{1}{2}{x}^2}{\mathrm{𝐻𝑒}}_{2n}\left(x\right)\cos \left(xy\right)dx={(-1)}^n\sqrt{\frac{1}{2}\pi }{y}^{2n}{\mathrm{e}}^{-\frac{1}{2}{y}^2},$$
ⓘ Symbols: ${\mathrm{𝐻𝑒}}_n\left(x\right)$: Hermite polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $y$: real variable, $n$: nonnegative integer and $x$: real variable Referenced by: §18.17(v) Permalink: http://dlmf.nist.gov/18.17.E23 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

18.17.24		$${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-x^2}{\mathrm{𝐻𝑒}}_{2n}\left(2x\right)\cos \left(xy\right)dx={(-1)}^n\frac{1}{2}\sqrt{\pi }{\mathrm{e}}^{-\frac{1}{4}{y}^2}{\mathrm{𝐻𝑒}}_{2n}\left(y\right).$$
ⓘ Symbols: ${\mathrm{𝐻𝑒}}_n\left(x\right)$: Hermite polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $y$: real variable, $n$: nonnegative integer and $x$: real variable Referenced by: §18.17(v) Permalink: http://dlmf.nist.gov/18.17.E24 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

18.17.25		$$\begin{array}{l}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{1}{2}{x}^2}{\mathrm{𝐻𝑒}}_n\left(x\right){\mathrm{𝐻𝑒}}_{n+2m}\left(x\right)\cos \left(xy\right)dx\\ ={(-1)}^m\sqrt{\frac{1}{2}\pi }n!y^{2m}{\mathrm{e}}^{-\frac{1}{2}{y}^2}{L}_n^{(2m)}\left(y^2\right),\end{array}$$
ⓘ Symbols: ${\mathrm{𝐻𝑒}}_n\left(x\right)$: Hermite polynomial, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $y$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Referenced by: §18.17(v) Permalink: http://dlmf.nist.gov/18.17.E25 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

18.17.26		$$\begin{array}{l}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{1}{2}{x}^2}{\mathrm{𝐻𝑒}}_n\left(x\right){\mathrm{𝐻𝑒}}_{n+2m+1}\left(x\right)\sin \left(xy\right)dx\\ ={(-1)}^m\sqrt{\frac{1}{2}\pi }n!y^{2m+1}{\mathrm{e}}^{-\frac{1}{2}{y}^2}{L}_n^{(2m+1)}\left(y^2\right).\end{array}$$
ⓘ Symbols: ${\mathrm{𝐻𝑒}}_n\left(x\right)$: Hermite polynomial, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $\sin z$: sine function, $y$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Referenced by: §18.17(v) Permalink: http://dlmf.nist.gov/18.17.E26 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

18.17.27		$${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{1}{2}{x}^2}{\mathrm{𝐻𝑒}}_{2n+1}\left(x\right)\sin \left(xy\right)dx={(-1)}^n\sqrt{\frac{1}{2}\pi }{y}^{2n+1}{\mathrm{e}}^{-\frac{1}{2}{y}^2},$$
ⓘ Symbols: ${\mathrm{𝐻𝑒}}_n\left(x\right)$: Hermite polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\sin z$: sine function, $y$: real variable, $n$: nonnegative integer and $x$: real variable Referenced by: §18.17(v) Permalink: http://dlmf.nist.gov/18.17.E27 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

18.17.28		$${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-x^2}{\mathrm{𝐻𝑒}}_{2n+1}\left(2x\right)\sin \left(xy\right)dx={(-1)}^n\frac{1}{2}\sqrt{\pi }{\mathrm{e}}^{-\frac{1}{4}{y}^2}{\mathrm{𝐻𝑒}}_{2n+1}\left(y\right).$$
ⓘ Symbols: ${\mathrm{𝐻𝑒}}_n\left(x\right)$: Hermite polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\sin z$: sine function, $y$: real variable, $n$: nonnegative integer and $x$: real variable Referenced by: §18.17(v) Permalink: http://dlmf.nist.gov/18.17.E28 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

18.17.28_5		$${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-x}{x}^{\alpha }{L}_n^{(\alpha )}\left(x\right){\mathrm{e}}^{\mathrm{i}xy}dx=\frac{\mathrm{\Gamma }\left(\alpha +n+1\right){(-\mathrm{i}y)}^n}{n!{\left(1-\mathrm{i}y\right)}^{\alpha +n+1}},$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\int$: integral, $y$: real variable, $n$: nonnegative integer and $x$: real variable Referenced by: §18.17(v), Erratum (V1.2.0) §18.17 Permalink: http://dlmf.nist.gov/18.17.E28_5 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

18.17.29		$$\begin{array}{l}{\int }_0^{\mathrm{\infty }}{x}^{2m}{\mathrm{e}}^{-\frac{1}{2}{x}^2}{L}_n^{(2m)}\left(x^2\right)\cos \left(xy\right)dx\\ ={(-1)}^m\sqrt{\frac{1}{2}\pi }\frac{1}{n!}{\mathrm{e}}^{-\frac{1}{2}{y}^2}{\mathrm{𝐻𝑒}}_n\left(y\right){\mathrm{𝐻𝑒}}_{n+2m}\left(y\right),\end{array}$$
ⓘ Symbols: ${\mathrm{𝐻𝑒}}_n\left(x\right)$: Hermite polynomial, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $y$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Referenced by: §18.17(v) Permalink: http://dlmf.nist.gov/18.17.E29 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

18.17.30		$${\int }_0^{\mathrm{\infty }}{x}^{2n}{\mathrm{e}}^{-\frac{1}{2}{x}^2}{L}_n^{(n-\frac{1}{2})}\left(\frac{1}{2}{x}^2\right)\cos \left(xy\right)dx=\sqrt{\frac{1}{2}\pi }{y}^{2n}{\mathrm{e}}^{-\frac{1}{2}{y}^2}{L}_n^{(n-\frac{1}{2})}\left(\frac{1}{2}{y}^2\right),$$
ⓘ Symbols: $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $y$: real variable, $n$: nonnegative integer and $x$: real variable Referenced by: §18.17(v) Permalink: http://dlmf.nist.gov/18.17.E30 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

18.17.31		$$\begin{array}{l}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-ax}{x}^{\nu -2n}{L}_{2n-1}^{(\nu -2n)}\left(ax\right)\cos \left(xy\right)dx\\ =i\frac{{(-1)}^n\mathrm{\Gamma }\left(\nu \right)}{2(2n-1)!}{y}^{2n-1}\left({(a+iy)}^{-\nu }-{(a-iy)}^{-\nu }\right),\end{array}$$
$\nu >2n-1$, $a>0$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\cos z$: cosine function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $y$: real variable, $n$: nonnegative integer and $x$: real variable Referenced by: §18.17(v) Permalink: http://dlmf.nist.gov/18.17.E31 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

18.17.32		$$\begin{array}{l}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-ax}{x}^{\nu -1-2n}{L}_{2n}^{(\nu -1-2n)}\left(ax\right)\cos \left(xy\right)dx\\ =\frac{{(-1)}^n\mathrm{\Gamma }\left(\nu \right)}{2(2n)!}{y}^{2n}\left({(a+iy)}^{-\nu }+{(a-iy)}^{-\nu }\right),\end{array}$$
$\nu >2n$, $a>0$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\cos z$: cosine function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $y$: real variable, $n$: nonnegative integer and $x$: real variable Referenced by: §18.17(v) Permalink: http://dlmf.nist.gov/18.17.E32 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18