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11: 12.13 Sums

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12.13.3 $$V(a,x+y)={\mathrm{e}}^{\frac{1}{2}xy+\frac{1}{4}{y}^2}\sum _{m=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{a-\frac{1}{2}}{m}\right)y^{m}V(a-m,x),$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\mathrm{e}$: base of natural logarithm, $V(a,z)$: parabolic cylinder function, $x$: real variable, $y$: real variable and $a$: real or complex parameter

Permalink:

http://dlmf.nist.gov/12.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §12.13(i), §12.13 and Ch.12

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12.13.4 $$V(a,x+y)={\mathrm{e}}^{-\frac{1}{2}xy-\frac{1}{4}{y}^2}\sum _{m=0}^{\mathrm{\infty }}\frac{y^m}{m!}V(a+m,x).$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $V(a,z)$: parabolic cylinder function, $x$: real variable, $y$: real variable and $a$: real or complex parameter

Referenced by:

§12.13(i)

Permalink:

http://dlmf.nist.gov/12.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.13(i), §12.13 and Ch.12

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12: 12.21 Software

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13: 12.4 Power-Series Expansions

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12.4.2 $$V(a,z)=V(a,0)u_1(a,z)+V^{\prime }(a,0)u_2(a,z),$$

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Symbols:

$V(a,z)$: parabolic cylinder function, $z$: complex variable, $a$: real or complex parameter, $u_1(a,z)$: solution and $u_2(a,z)$: solution

Referenced by:

§12.20

Permalink:

http://dlmf.nist.gov/12.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §12.4 and Ch.12

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14: 12.19 Tables

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Abramowitz and Stegun (1964, Chapter 19) includes $U(a,x)$ and $V(a,x)$ for $\pm a=0(.1)1(.5)5$, $x=0(.1)5$, 5S; $W(a,\pm x)$ for $\pm a=0(.1)1(1)5$, $x=0(.1)5$, 4-5D or 4-5S.

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Zhang and Jin (1996, pp. 455–473) includes $U(\pm n-\frac{1}{2},x)$, $V(\pm n-\frac{1}{2},x)$, $U(\pm \nu -\frac{1}{2},x)$, $V(\pm \nu -\frac{1}{2},x)$, and derivatives, $\nu =n+\frac{1}{2}$, $n=0(1)10(10)30$, $x=0.5,1,5,10,30,50$, 8S; $W(a,\pm x)$, $W(-a,\pm x)$, and derivatives, $a=h(1)5+h$, $x=0.5,1$ and $a=h(1)5+h$, $x=5$, $h=0,0.5$, 8S. Also, first zeros of $U(a,x)$, $V(a,x)$, and of derivatives, $a=-6(.5)-1$, 6D; first three zeros of $W(a,-x)$ and of derivative, $a=0(.5)4$, 6D; first three zeros of $W(-a,\pm x)$ and of derivative, $a=0.5(.5)5.5$, 6D; real and imaginary parts of $U(a,z)$, $a=-1.5(1)1.5$, $z=x+\mathrm{i}y$, $x=0.5,1,5,10$, $y=0(.5)10$, 8S.

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15: 12.20 Approximations

… ►As special cases of these results a Chebyshev-series expansion for $U(a,x)$ valid when follows from (12.7.14), and Chebyshev-series expansions for $U(a,x)$ and $V(a,x)$ valid when $0\le x\le \lambda$ follow from (12.4.1), (12.4.2), (12.7.12), and (12.7.13). …

16: 35.2 Laplace Transform

… ►Assume that ${\int }_{𝓢}\left|g(𝐔+\mathrm{i}𝐕)\right|d𝐕$ converges, and also that its limit as $𝐔\to \mathrm{\infty }$ is $0$. …where the integral is taken over all $𝐙=𝐔+\mathrm{i}𝐕$ such that $𝐔>{𝐗}_0$ and $𝐕$ ranges over $𝓢$. …

17: 12.17 Physical Applications

… ►Setting $w=U(\xi )V(\eta )W(\zeta )$ and separating variables, we obtain … ►

$$\frac{d^{2}V}{{d\eta }^2}+\left(\sigma {\eta }^2-\lambda \right)V=0,$$

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18: 10.69 Uniform Asymptotic Expansions for Large Order

… ►Let $U_k(p)$ and $V_k(p)$ be the polynomials defined in §10.41(ii), and … ►

10.69.4 $${\mathrm{ber}}_{\nu }^{\prime }\left(\nu x\right)+\mathrm{i}{\mathrm{bei}}_{\nu }^{\prime }\left(\nu x\right)\sim \frac{{\mathrm{e}}^{\nu \xi }}{x}{\left(\frac{\xi }{2\pi \nu }\right)}^{1/2}{\left(\frac{x{\mathrm{e}}^{3\pi \mathrm{i}/4}}{1+\xi }\right)}^{\nu }\sum _{k=0}^{\mathrm{\infty }}\frac{V_k({\xi }^{-1})}{{\nu }^k},$$

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Symbols:

${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $x$: real variable, $\nu$: complex parameter, $V_k(p)$: polynomial coefficient and $\xi$

A&S Ref:

9.10.39

Referenced by:

§10.69

Permalink:

http://dlmf.nist.gov/10.69.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.69 and Ch.10

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10.69.5 $${\ker }_{\nu }^{\prime }\left(\nu x\right)+\mathrm{i}{\mathrm{kei}}_{\nu }^{\prime }\left(\nu x\right)\sim -\frac{{\mathrm{e}}^{-\nu \xi }}{x}{\left(\frac{\pi \xi }{2\nu }\right)}^{1/2}{\left(\frac{x{\mathrm{e}}^{3\pi \mathrm{i}/4}}{1+\xi }\right)}^{-\nu }\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{V_k({\xi }^{-1})}{{\nu }^k},$$

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Symbols:

${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $x$: real variable, $\nu$: complex parameter, $V_k(p)$: polynomial coefficient and $\xi$

A&S Ref:

9.10.40

Permalink:

http://dlmf.nist.gov/10.69.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.69 and Ch.10

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19: 12.9 Asymptotic Expansions for Large Variable

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12.9.2 $$V(a,z)\sim \sqrt{\frac{2}{\pi }}{\mathrm{e}}^{\frac{1}{4}{z}^2}{z}^{a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-a\right)}_{2s}}{s!{(2z^2)}^s},$$ .

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $V(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable, $s$: nonnegative integer, $a$: real or complex parameter and $\delta$: arbitrary small positive constant

A&S Ref:

19.8.2 (modification of)

Referenced by:

§12.9(i), §12.9(ii)

Permalink:

http://dlmf.nist.gov/12.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §12.9(i), §12.9 and Ch.12

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12.9.4 $$V(a,z)\sim \sqrt{\frac{2}{\pi }}{\mathrm{e}}^{\frac{1}{4}{z}^2}{z}^{a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-a\right)}_{2s}}{s!{(2z^2)}^s}\pm \frac{\mathrm{i}}{\mathrm{\Gamma }\left(\frac{1}{2}-a\right)}{\mathrm{e}}^{-\frac{1}{4}{z}^2}{z}^{-a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(\frac{1}{2}+a\right)}_{2s}}{s!{(2z^2)}^s},$$ $-\frac{1}{4}\pi +\delta \le \pm \mathrm{ph}z\le \frac{3}{4}\pi -\delta$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $V(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable, $s$: nonnegative integer, $a$: real or complex parameter and $\delta$: arbitrary small positive constant

Referenced by:

§12.9(i)

Permalink:

http://dlmf.nist.gov/12.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.9(i), §12.9 and Ch.12

►To obtain approximations for $U(a,-z)$ and $V(a,-z)$ as $z\to \mathrm{\infty }$ combine the results above with (12.2.15) and (12.2.16). …

20: 10.41 Asymptotic Expansions for Large Order

… ►Also, $U_k(p)$ and $V_k(p)$ are polynomials in $p$ of degree $3k$, given by $U_0(p)=V_0(p)=1$, and … ►

$V_1(p)=\frac{1}{24}(-9p+7p^3),$

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$V_2(p)=\frac{1}{1152}(-135p^2+594p^4-455p^6),$

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$V_3(p)=\frac{1}{\mathrm{4\hspace{0.33em}14720}}(-42525p^3+\mathrm{4\hspace{0.33em}51737}{p}^5-\mathrm{8\hspace{0.33em}83575}{p}^7+\mathrm{4\hspace{0.33em}75475}{p}^9).$

… ►For numerical tables of $\eta =\eta (z)$ and the coefficients $U_k(p)$, $V_k(p)$, see Olver (1962, pp. 43–51). …