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21: 12.8 Recurrence Relations and Derivatives

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§12.8(i) Recurrence Relations

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12.8.1 $$zU(a,z)-U(a-1,z)+(a+\frac{1}{2})U(a+1,z)=0,$$

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Symbols:

$U(a,z)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.6.4

Referenced by:

§12.8(i)

Permalink:

http://dlmf.nist.gov/12.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §12.8(i), §12.8 and Ch.12

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12.8.2 $$U^{\prime }(a,z)+\frac{1}{2}zU(a,z)+(a+\frac{1}{2})U(a+1,z)=0,$$

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Symbols:

$U(a,z)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.6.1

Permalink:

http://dlmf.nist.gov/12.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §12.8(i), §12.8 and Ch.12

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12.8.3 $$U^{\prime }(a,z)-\frac{1}{2}zU(a,z)+U(a-1,z)=0,$$

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Symbols:

$U(a,z)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.6.2

Permalink:

http://dlmf.nist.gov/12.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §12.8(i), §12.8 and Ch.12

… ►

§12.8(ii) Derivatives

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22: 12.12 Integrals

§12.12 Integrals

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Nicholson-type Integral

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12.12.4 $${(U(a,z))}^2+{(\overline{U}(a,z))}^2=\frac{2^{\frac{3}{2}}}{\pi }\mathrm{\Gamma }\left(\frac{1}{2}-a\right){\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{2at+\frac{1}{2}{z}^2\tanh t}}{\sqrt{\sinh \left(2t\right)}}dt,$$ .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\tanh z$: hyperbolic tangent function, $\int$: integral, $U(a,z)$: parabolic cylinder function, $\overline{U}(a,x)$: parabolic cylinder function, $\mathrm{\Re }$: real part, $z$: complex variable and $a$: real or complex parameter

Notes:

See Durand (1975).

Referenced by:

§12.12

Permalink:

http://dlmf.nist.gov/12.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.12, §12.12 and Ch.12

… ► ►See also Barr (1968) and Lowdon (1970).

23: 10.43 Integrals

… ►Let ${𝒵}_{\nu }\left(z\right)$ be defined as in §10.25(ii). … ►

$$\int z^{\nu +1}{𝒵}_{\nu }\left(z\right)dz=z^{\nu +1}{𝒵}_{\nu +1}\left(z\right),$$

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$$\int z^{-\nu +1}{𝒵}_{\nu }\left(z\right)dz=z^{-\nu +1}{𝒵}_{\nu -1}\left(z\right).$$

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$$\int {\mathrm{e}}^{\pm z}{z}^{\nu }{𝒵}_{\nu }\left(z\right)dz=\frac{{\mathrm{e}}^{\pm z}{z}^{\nu +1}}{2\nu +1}\left({𝒵}_{\nu }\left(z\right)\mp {𝒵}_{\nu +1}\left(z\right)\right),$$ $\nu \ne -\frac{1}{2}$,

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$$\int {\mathrm{e}}^{\pm z}{z}^{-\nu }{𝒵}_{\nu }\left(z\right)dz=\frac{{\mathrm{e}}^{\pm z}{z}^{-\nu +1}}{1-2\nu }\left({𝒵}_{\nu }\left(z\right)\mp {𝒵}_{\nu -1}\left(z\right)\right),$$ $\nu \ne \frac{1}{2}$.

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24: 10.22 Integrals

… ►In this subsection ${𝒞}_{\nu }\left(z\right)$ and ${𝒟}_{\mu }(z)$ denote cylinder functions(§10.2(ii)) of orders $\nu$ and $\mu$, respectively, not necessarily distinct. ►

${\displaystyle \int z^{\nu +1}{𝒞}_{\nu }\left(z\right)dz}=z^{\nu +1}{𝒞}_{\nu +1}\left(z\right),$

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${\displaystyle \int z^{-\nu +1}{𝒞}_{\nu }\left(z\right)dz}=-z^{-\nu +1}{𝒞}_{\nu -1}\left(z\right).$

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$$\int {\mathrm{e}}^{\mathrm{i}z}{z}^{\nu }{𝒞}_{\nu }\left(z\right)dz=\frac{{\mathrm{e}}^{\mathrm{i}z}{z}^{\nu +1}}{2\nu +1}({𝒞}_{\nu }\left(z\right)-\mathrm{i}{𝒞}_{\nu +1}\left(z\right)),$$ $\nu \ne -\frac{1}{2}$,

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10.22.5 $$\int z{𝒞}_{\mu }\left(az\right){𝒟}_{\mu }(az)dz=\frac{1}{4}{z}^2\left(2{𝒞}_{\mu }\left(az\right){𝒟}_{\mu }(az)-{𝒞}_{\mu -1}\left(az\right){𝒟}_{\mu +1}(az)-{𝒞}_{\mu +1}\left(az\right){𝒟}_{\mu -1}(az)\right),$$

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Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $dx$: differential of $x$ , $\int$: integral, $z$: complex variable and ${𝒟}_{\nu }(z)$: cylinder function

Referenced by:

§10.22(ii)

Permalink:

http://dlmf.nist.gov/10.22.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.22(i), §10.22(i), §10.22 and Ch.10

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25: 12.13 Sums

§12.13 Sums

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§12.13(i) Addition Theorems

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12.13.1 $$U(a,x+y)={\mathrm{e}}^{\frac{1}{2}xy+\frac{1}{4}{y}^2}\sum _{m=0}^{\mathrm{\infty }}\frac{{(-y)}^m}{m!}U(a-m,x),$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $U(a,z)$: 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.E1

Encodings:

pMML, png, TeX

See also:

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

►

12.13.2 $$U(a,x+y)={\mathrm{e}}^{-\frac{1}{2}xy-\frac{1}{4}{y}^2}\sum _{m=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{-a-\frac{1}{2}}{m}\right)y^{m}U(a+m,x),$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/12.13.E2

Encodings:

pMML, png, TeX

See also:

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

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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

…

26: 10.2 Definitions

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§10.2(i) Bessel’s Equation

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Cylinder Functions

►The notation ${𝒞}_{\nu }\left(z\right)$ denotes $J_{\nu }\left(z\right)$, $Y_{\nu }\left(z\right)$, $H_{\nu }^{(1)}\left(z\right)$, $H_{\nu }^{(2)}\left(z\right)$, or any nontrivial linear combination of these functions, the coefficients in which are independent of $z$ and $\nu$. …

27: 10.21 Zeros

… ►The positive zeros of any two real distinct cylinder functions of the same order are interlaced, as are the positive zeros of any real cylinder function ${𝒞}_{\nu }\left(z\right)$ and the contiguous function ${𝒞}_{\nu +1}\left(z\right)$. … ►

10.21.5 $${𝒞}_{\nu }^{\prime }\left({\rho }_{\nu }\right)={𝒞}_{\nu -1}\left({\rho }_{\nu }\right)=-{𝒞}_{\nu +1}\left({\rho }_{\nu }\right).$$

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Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $\nu$: complex parameter and ${\rho }_{\nu }(t)$: zero of cylinder function

A&S Ref:

9.5.4

Referenced by:

§10.21(ii)

Permalink:

http://dlmf.nist.gov/10.21.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.21(ii), §10.21 and Ch.10

►If ${\sigma }_{\nu }$ is a zero of ${𝒞}_{\nu }^{\prime }\left(z\right)$, then … ►Any positive zero $c$ of the cylinder function ${𝒞}_{\nu }\left(x\right)$ and any positive zero $c^{\prime }$ of ${𝒞}_{\nu }^{\prime }\left(x\right)$ such that $c^{\prime }>|\nu |$ are definable as continuous and increasing functions of $\nu$: … ► …

28: 12.9 Asymptotic Expansions for Large Variable

§12.9 Asymptotic Expansions for Large Variable

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12.9.1 $$U(a,z)\sim {\mathrm{e}}^{-\frac{1}{4}{z}^2}{z}^{-a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(\frac{1}{2}+a\right)}_{2s}}{s!{(2z^2)}^s},$$ ,

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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 $n!$), $U(a,z)$: 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:

19.8.1 (modification of)

Referenced by:

§12.9(i), §12.9(ii)

Permalink:

http://dlmf.nist.gov/12.9.E1

Encodings:

pMML, png, TeX

See also:

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

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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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12.9.3 $$U(a,z)\sim {\mathrm{e}}^{-\frac{1}{4}{z}^2}{z}^{-a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(\frac{1}{2}+a\right)}_{2s}}{s!{(2z^2)}^s}\pm \mathrm{i}\frac{\sqrt{2\pi }}{\mathrm{\Gamma }\left(\frac{1}{2}+a\right)}{\mathrm{e}}^{\mp \mathrm{i}\pi a}{\mathrm{e}}^{\frac{1}{4}{z}^2}{z}^{a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-a\right)}_{2s}}{s!{(2z^2)}^s},$$ $\frac{1}{4}\pi +\delta \le \pm \mathrm{ph}z\le \frac{5}{4}\pi -\delta$ ,

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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, $U(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable, $s$: nonnegative integer, $a$: real or complex parameter and $\delta$: arbitrary small positive constant

Permalink:

http://dlmf.nist.gov/12.9.E3

Encodings:

pMML, png, TeX

See also:

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

… ►

§12.9(ii) Bounds and Re-Expansions for the Remainder Terms

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29: 12.5 Integral Representations

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§12.5(i) Integrals Along the Real Line

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12.5.1 $$U(a,z)=\frac{{\mathrm{e}}^{-\frac{1}{4}{z}^2}}{\mathrm{\Gamma }\left(\frac{1}{2}+a\right)}{\int }_0^{\mathrm{\infty }}{t}^{a-\frac{1}{2}}{\mathrm{e}}^{-\frac{1}{2}{t}^2-zt}dt,$$ $\mathrm{\Re }a>-\frac{1}{2}$ ,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $U(a,z)$: parabolic cylinder function, $\mathrm{\Re }$: real part, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.5.3 (modification of)

Referenced by:

§12.5(i), §12.8(ii), §36.2(ii)

Permalink:

http://dlmf.nist.gov/12.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §12.5(i), §12.5 and Ch.12

… ►

§12.5(ii) Contour Integrals

… ►

§12.5(iii) Mellin–Barnes Integrals

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§12.5(iv) Compendia

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30: 29.16 Asymptotic Expansions

… ►The approximating functions are exponential, trigonometric, and parabolic cylinder functions.