cylinder functions

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11: 12.18 Methods of Computation

§12.18 Methods of Computation

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12: 12.15 Generalized Parabolic Cylinder Functions

§12.15 Generalized Parabolic Cylinder Functions

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13: 12.20 Approximations

§12.20 Approximations

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14: 10.66 Expansions in Series of Bessel Functions

§10.66 Expansions in Series of Bessel Functions

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10.66.1 $${\mathrm{ber}}_{\nu }x+\mathrm{i}{\mathrm{bei}}_{\nu }x=\sum _{k=0}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{(3\nu +k)\pi \mathrm{i}/4}{x}^k{J}_{\nu +k}\left(x\right)}{2^{k/2}k!}=\sum _{k=0}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{(3\nu +3k)\pi \mathrm{i}/4}{x}^k{I}_{\nu +k}\left(x\right)}{2^{k/2}k!}.$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, ${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, $\pi$: 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, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $k$: nonnegative integer, $x$: real variable and $\nu$: complex parameter

A&S Ref:

9.9.32

Referenced by:

§10.66

Permalink:

http://dlmf.nist.gov/10.66.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.66 and Ch.10

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15: 10.23 Sums

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§10.23(i) Multiplication Theorem

… ►If $𝒞=J$ and the upper signs are taken, then the restriction on $\lambda$ is unnecessary. ►

§10.23(ii) Addition Theorems

… ►The restriction is unnecessary when $𝒞=J$ and $\nu$ is an integer. … ►The restriction is unnecessary in (10.23.7) when $𝒞=J$ and $\nu$ is an integer, and in (10.23.8) when $𝒞=J$. …

16: 12.17 Physical Applications

§12.17 Physical Applications

… ►By using instead coordinates of the parabolic cylinder $\xi ,\eta ,\zeta$, defined by … ► … ►Dean (1966) describes the role of PCFs in quantum mechanical systems closely related to the one-dimensional harmonic oscillator. … ►Lastly, parabolic cylinder functions arise in the description of ultra cold atoms in harmonic trapping potentials; see Busch et al. (1998) and Edwards et al. (1999).

17: 12.7 Relations to Other Functions

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§12.7(i) Hermite Polynomials

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12.7.1 $$U(-\frac{1}{2},z)=D_0\left(z\right)={\mathrm{e}}^{-\frac{1}{4}{z}^2},$$

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Symbols:

$D_{\nu }\left(z\right)$: parabolic cylinder function, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/12.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §12.7(i), §12.7 and Ch.12

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§12.7(ii) Error Functions, Dawson’s Integral, and Probability Function

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§12.7(iii) Modified Bessel Functions

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§12.7(iv) Confluent Hypergeometric Functions

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18: 10.25 Definitions

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Symbol ${𝒵}_{\nu }\left(z\right)$

►Corresponding to the symbol ${𝒞}_{\nu }$ introduced in §10.2(ii), we sometimes use ${𝒵}_{\nu }\left(z\right)$ to denote $I_{\nu }\left(z\right)$, ${\mathrm{e}}^{\nu \pi \mathrm{i}}{K}_{\nu }\left(z\right)$, or any nontrivial linear combination of these functions, the coefficients in which are independent of $z$ and $\nu$. …

19: 12.2 Differential Equations

§12.2 Differential Equations

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§12.2(i) Introduction

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§12.2(iii) Wronskians

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§12.2(iv) Reflection Formulas

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§12.2(v) Connection Formulas

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20: 12.3 Graphics

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§12.3(i) Real Variables

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Figure 12.3.8: $V(a,x)$, $-2.5\le a\le 2.5$, $-2.5\le x\le 2.5$. Magnify 3D Help

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§12.3(ii) Complex Variables

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