12 Parabolic Cylinder Functions Properties 12.11 Zeros 12.13 Sums

§12.12 Integrals

Notes:: See Erdélyi et al. (1953b, v. 2, Chapter 8). For (12.12.4) see Durand (1975).
Keywords:: integrals, parabolic cylinder functions
Permalink:: http://dlmf.nist.gov/12.12

12.12.1 $\int_{0}^{\infty}e^{{-\frac{1}{4}t^{2}}}t^{{\mu-1}}\mathop{U\/}\nolimits\!% \left(a,t\right)dt=\frac{\sqrt{\pi}2^{{-\frac{1}{2}(\mu+a+\frac{1}{2})}}% \mathop{\Gamma\/}\nolimits\!\left(\mu\right)}{\mathop{\Gamma\/}\nolimits\!% \left(\frac{1}{2}(\mu+a+\frac{3}{2})\right)},$ $\realpart{\mu}>0$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : differential of , : base of exponential function, $\int$ : integral, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $\realpart{}$ : real part and : real or complex parameter
Permalink:: http://dlmf.nist.gov/12.12.E1
Encodings:: TeX, pMML, png

12.12.2 $\int_{0}^{\infty}e^{{-\frac{3}{4}t^{2}}}t^{{-a-\frac{3}{2}}}\mathop{U\/}% \nolimits\!\left(a,t\right)dt=2^{{\frac{1}{4}+\frac{1}{2}a}}\mathop{\Gamma\/}% \nolimits\!\left(-a-\tfrac{1}{2}\right)\mathop{\cos\/}\nolimits\!\left((\tfrac% {1}{4}a+\tfrac{1}{8})\pi\right),$ $\realpart{a}<-\tfrac{1}{2}$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , : base of exponential function, $\int$ : integral, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $\realpart{}$ : real part and : real or complex parameter
Permalink:: http://dlmf.nist.gov/12.12.E2
Encodings:: TeX, pMML, png

12.12.3 $\int_{0}^{\infty}e^{{-\frac{1}{4}t^{2}}}t^{{-a-\frac{1}{2}}}(x^{2}+t^{2})^{{-1% }}\mathop{U\/}\nolimits\!\left(a,t\right)dt=\sqrt{\pi/2}\mathop{\Gamma\/}% \nolimits\!\left(\tfrac{1}{2}-a\right)x^{{-a-\frac{3}{2}}}e^{{\frac{1}{4}x^{2}% }}\mathop{U\/}\nolimits\!\left(-a,x\right),$ $\realpart{a}<\tfrac{1}{2},x>0$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : differential of , : base of exponential function, $\int$ : integral, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $\realpart{}$ : real part, : real variable and : real or complex parameter
Permalink:: http://dlmf.nist.gov/12.12.E3
Encodings:: TeX, pMML, png

¶ Nicholson-type Integral

Keywords:: Nicholson-type integral, integrals, parabolic cylinder functions

12.12.4 $(\mathop{U\/}\nolimits\!\left(a,z\right))^{2}+(\mathop{\overline{U}\/}% \nolimits\!\left(a,z\right))^{2}=\frac{2^{{\frac{3}{2}}}}{\pi}\mathop{\Gamma\/% }\nolimits\!\left(\tfrac{1}{2}-a\right)\int_{0}^{\infty}\frac{e^{{2at+\frac{1}% {2}z^{2}\mathop{\tanh\/}\nolimits t}}}{\sqrt{\mathop{\sinh\/}\nolimits(2t)}}dt,$ $\realpart{a}<\tfrac{1}{2}$ .

Notes:: See Durand (1975).
Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : differential of , : base of exponential function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\mathop{\tanh\/}\nolimits z$ : hyperbolic tangent function, $\int$ : integral, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $\mathop{\overline{U}\/}\nolimits\!\left(a,x\right)$ : parabolic cylinder function, $\realpart{}$ : real part, : complex variable and : real or complex parameter
Referenced by:: §12.12
Permalink:: http://dlmf.nist.gov/12.12.E4
Encodings:: TeX, pMML, png

When is real the left-hand side equals $(F(a,x))^{2}$ ; compare (12.2.22).

For further integrals see §§13.10, 13.23, and use (12.7.14).

For compendia of integrals see Erdélyi et al. (1953b, v. 2, pp. 121–122), Erdélyi et al. (1954a, b, v. 1, pp. 60–61, 115, 210–211, and 336; v. 2, pp. 76–80, 115, 151, 171, and 395–398), Gradshteyn and Ryzhik (2000, §7.7), Magnus et al. (1966, pp. 330–331), Marichev (1983, pp. 190–191), Oberhettinger (1974, pp. 144–145), Oberhettinger (1990, pp. 106–108 and 192), Oberhettinger and Badii (1973, pp. 181–185), Prudnikov et al. (1986b, pp. 36–37, 155–168, 243–246, 289–290, 327–328, 419–420, and 619), Prudnikov et al. (1992a, §3.11), and Prudnikov et al. (1992b, §3.11).