§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) 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) 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) 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

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) 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) 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).

18.17.7		$${\left(P_n\left(x\right)\right)}^2+4{\pi }^{-2}{\left({\mathsf{Q}}_n\left(x\right)\right)}^2=4{\pi }^{-2}{\int }_1^{\mathrm{\infty }}{Q}_n\left(x^2+(1-x^2)t\right){(t^2-1)}^{-\frac{1}{2}}dt,$$
.
ⓘ 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, ${\mathsf{Q}}_{\nu }\left(x\right)={\mathsf{Q}}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: Legendre function of the second kind, $n$: 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 ${\mathsf{Q}}_n\left(x\right)$ 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, $n$: 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) 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.

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 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) 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

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 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) 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

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.

18.17.17		$${\int }_0^1{(1-x^2)}^{\lambda -\frac{1}{2}}{C}_{2n}^{(\lambda )}\left(x\right)\cos \left(xy\right)dx=\frac{{(-1)}^n\pi \mathrm{\Gamma }\left(2n+2\lambda \right)J_{\lambda +2n}\left(y\right)}{(2n)!\mathrm{\Gamma }\left(\lambda \right){(2y)}^{\lambda }},$$
ⓘ 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

18.17.22		$$\frac{1}{2\sqrt{\pi }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{1}{4}{x}^2}{\mathit{He}}_n\left(x\right){\mathrm{e}}^{\frac{1}{2}\mathrm{i}xy}dx={\mathrm{i}}^n{\mathrm{e}}^{-\frac{1}{4}{y}^2}{\mathit{He}}_n\left(y\right),$$
ⓘ Symbols: ${\mathit{He}}_n\left(x\right)$: 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		$${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{1}{2}{x}^2}{\mathit{He}}_{2n}\left(x\right)\cos \left(xy\right)dx={(-1)}^n\sqrt{\frac{1}{2}\pi }{y}^{2n}{\mathrm{e}}^{-\frac{1}{2}{y}^2},$$
ⓘ Symbols: ${\mathit{He}}_n\left(x\right)$: 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		$${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-x^2}{\mathit{He}}_{2n}\left(2x\right)\cos \left(xy\right)dx={(-1)}^n\frac{1}{2}\sqrt{\pi }{\mathrm{e}}^{-\frac{1}{4}{y}^2}{\mathit{He}}_{2n}\left(y\right).$$
ⓘ Symbols: ${\mathit{He}}_n\left(x\right)$: 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		$$\begin{array}{l}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{1}{2}{x}^2}{\mathit{He}}_n\left(x\right){\mathit{He}}_{n+2m}\left(x\right)\cos \left(xy\right)dx\\ ={(-1)}^m\sqrt{\frac{1}{2}\pi }n!y^{2m}{\mathrm{e}}^{-\frac{1}{2}{y}^2}{L}_n^{(2m)}\left(y^2\right),\end{array}$$
ⓘ Symbols: ${\mathit{He}}_n\left(x\right)$: 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		$$\begin{array}{l}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{1}{2}{x}^2}{\mathit{He}}_n\left(x\right){\mathit{He}}_{n+2m+1}\left(x\right)\sin \left(xy\right)dx\\ ={(-1)}^m\sqrt{\frac{1}{2}\pi }n!y^{2m+1}{\mathrm{e}}^{-\frac{1}{2}{y}^2}{L}_n^{(2m+1)}\left(y^2\right).\end{array}$$
ⓘ Symbols: ${\mathit{He}}_n\left(x\right)$: 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		$${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{1}{2}{x}^2}{\mathit{He}}_{2n+1}\left(x\right)\sin \left(xy\right)dx={(-1)}^n\sqrt{\frac{1}{2}\pi }{y}^{2n+1}{\mathrm{e}}^{-\frac{1}{2}{y}^2},$$
ⓘ Symbols: ${\mathit{He}}_n\left(x\right)$: 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		$${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-x^2}{\mathit{He}}_{2n+1}\left(2x\right)\sin \left(xy\right)dx={(-1)}^n\frac{1}{2}\sqrt{\pi }{\mathrm{e}}^{-\frac{1}{4}{y}^2}{\mathit{He}}_{2n+1}\left(y\right).$$
ⓘ Symbols: ${\mathit{He}}_n\left(x\right)$: 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

18.17.29		$$\begin{array}{l}{\int }_0^{\mathrm{\infty }}{x}^{2m}{\mathrm{e}}^{-\frac{1}{2}{x}^2}{L}_n^{(2m)}\left(x^2\right)\cos \left(xy\right)dx\\ ={(-1)}^m\sqrt{\frac{1}{2}\pi }\frac{1}{n!}{\mathrm{e}}^{-\frac{1}{2}{y}^2}{\mathit{He}}_n\left(y\right){\mathit{He}}_{n+2m}\left(y\right).\end{array}$$
ⓘ Symbols: ${\mathit{He}}_n\left(x\right)$: 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		$$\begin{array}{l}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-ax}{x}^{\nu -2n}{L}_{2n-1}^{(\nu -2n)}\left(ax\right)\cos \left(xy\right)dx\\ =\mathrm{i}\frac{{(-1)}^n\mathrm{\Gamma }\left(\nu \right)}{2(2n-1)!}{y}^{2n-1}\left({(a+\mathrm{i}y)}^{-\nu }-{(a-\mathrm{i}y)}^{-\nu }\right),\end{array}$$
$\nu >2n-1$, $a>0$,
ⓘ 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 $n!$), $\mathrm{i}$: 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.E31 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

18.17.32		$$\begin{array}{l}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-ax}{x}^{\nu -1-2n}{L}_{2n}^{(\nu -1-2n)}\left(ax\right)\cos \left(xy\right)dx\\ =\frac{{(-1)}^n\mathrm{\Gamma }\left(\nu \right)}{2(2n)!}{y}^{2n}\left({(a+\mathrm{i}y)}^{-\nu }+{(a-\mathrm{i}y)}^{-\nu }\right),\end{array}$$
$\nu >2n$, $a>0$.
ⓘ 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 $n!$), $\mathrm{i}$: 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.E32 Encodings: TeX, pMML, png See also: Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

18.17.33		$$\begin{array}{l}{\int }_{-1}^1{\mathrm{e}}^{-(x+1)z}{P}_n^{(\alpha ,\beta )}\left(x\right){(1-x)}^{\alpha }{(1+x)}^{\beta }dx\\ =\frac{{(-1)}^n{2}^{\alpha +\beta +n+1}\mathrm{\Gamma }\left(\alpha +n+1\right)\mathrm{\Gamma }\left(\beta +n+1\right)}{\mathrm{\Gamma }\left(\alpha +\beta +2n+2\right)n!}{z}^n{}_{1}F_1(\genfrac{}{}{0pt}{}{\beta +n+1}{\alpha +\beta +2n+2};-2z),\end{array}$$
$z\in ℂ$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma 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, $ℂ$: complex plane, $dx$: differential of $x$, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $z$: complex variable, $n$: nonnegative integer and $x$: real variable Referenced by: §18.17(vi) Permalink: http://dlmf.nist.gov/18.17.E33 Encodings: TeX, pMML, png See also: Annotations for §18.17(vi), §18.17(vi), §18.17 and Ch.18

For the confluent hypergeometric function ${}_{1}F_1$ see (16.2.1) and Chapter 13.