cylinder functions

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1: 12.1 Special Notation

… ►Unless otherwise noted, primes indicate derivatives with respect to the variable, and fractional powers take their principal values. ►The main functions treated in this chapter are the parabolic cylinder functions (PCFs), also known as Weber parabolic cylinder functions: $U(a,z)$, $V(a,z)$, $\overline{U}(a,z)$, and $W(a,z)$. …An older notation, due to Whittaker (1902), for $U(a,z)$ is $D_{\nu }\left(z\right)$. …

2: 12.14 The Function $W(a,x)$

§12.14 The Function $W(a,x)$

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§12.14(iii) Graphs

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§12.14(iv) Connection Formula

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Bessel Functions

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3: 10.29 Recurrence Relations and Derivatives

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

${𝒵}_{\nu -1}\left(z\right)-{𝒵}_{\nu +1}\left(z\right)=(2\nu /z){𝒵}_{\nu }\left(z\right),$

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${𝒵}_{\nu -1}\left(z\right)+{𝒵}_{\nu +1}\left(z\right)=2{𝒵}_{\nu }^{\prime }\left(z\right).$

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${𝒵}_{\nu }^{\prime }\left(z\right)={𝒵}_{\nu -1}\left(z\right)-(\nu /z){𝒵}_{\nu }\left(z\right),$

… ►For results on modified quotients of the form $z{𝒵}_{\nu \pm 1}\left(z\right)/{𝒵}_{\nu }\left(z\right)$ see Onoe (1955) and Onoe (1956). …

4: 10.36 Other Differential Equations

… ►The quantity ${\lambda }^2$ in (10.13.1)–(10.13.6) and (10.13.8) can be replaced by $-{\lambda }^2$ if at the same time the symbol $𝒞$ in the given solutions is replaced by $𝒵$. … ►

10.36.1 $$z^2(z^2+{\nu }^2)w^{\prime \prime }+z(z^2+3{\nu }^2)w^{\prime }-\left({(z^2+{\nu }^2)}^2+z^2-{\nu }^2\right)w=0,$$ $w={𝒵}_{\nu }^{\prime }\left(z\right)$,

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Symbols:

${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.36.E1

Encodings:

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See also:

Annotations for §10.36 and Ch.10

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10.36.2 $$z^2{w}^{\prime \prime }+z(1\pm 2z)w^{\prime }+(\pm z-{\nu }^2)w=0,$$ $w={\mathrm{e}}^{\mp z}{𝒵}_{\nu }\left(z\right)$.

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Symbols:

$\mathrm{e}$: base of natural logarithm, ${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.41

Permalink:

http://dlmf.nist.gov/10.36.E2

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Annotations for §10.36 and Ch.10

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5: 12.16 Mathematical Applications

§12.16 Mathematical Applications

… ►For examples see §§13.20(iii), 13.20(iv), 14.15(v), and 14.26. … ► … ►

6: 10.13 Other Differential Equations

§10.13 Other Differential Equations

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10.13.1 $$w^{\prime \prime }+\left({\lambda }^2-\frac{{\nu }^2-\frac{1}{4}}{z^2}\right)w=0,$$ $w=z^{\frac{1}{2}}{𝒞}_{\nu }\left(\lambda z\right)$,

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Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.49

Referenced by:

§1.18(vi), §10.36

Permalink:

http://dlmf.nist.gov/10.13.E1

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Annotations for §10.13 and Ch.10

… ►In (10.13.9)–(10.13.11) ${𝒞}_{\nu }\left(z\right)$, ${𝒟}_{\mu }(z)$ are any cylinder functions of orders $\nu ,\mu$, respectively, and $\mathit{\vartheta }=z(d/dz)$. ►

10.13.9 $$z^2{w}^{\prime \prime \prime }+3z{w}^{\prime \prime }+(4z^2+1-4{\nu }^2)w^{\prime }+4zw=0,$$ $w={𝒞}_{\nu }\left(z\right){𝒟}_{\nu }(z)$,

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Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $z$: complex variable, $\nu$: complex parameter and ${𝒟}_{\nu }(z)$: cylinder function

A&S Ref:

9.1.58 (modified)

Referenced by:

§10.13, §10.36

Permalink:

http://dlmf.nist.gov/10.13.E9

Encodings:

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Annotations for §10.13 and Ch.10

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10.13.10 $$z^3{w}^{\prime \prime \prime }+z(4z^2+1-4{\nu }^2)w^{\prime }+(4{\nu }^2-1)w=0,$$ $w=z{𝒞}_{\nu }\left(z\right){𝒟}_{\nu }(z)$,

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Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $z$: complex variable, $\nu$: complex parameter and ${𝒟}_{\nu }(z)$: cylinder function

A&S Ref:

9.1.59

Permalink:

http://dlmf.nist.gov/10.13.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §10.13 and Ch.10

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7: 10.6 Recurrence Relations and Derivatives

… ►With ${𝒞}_{\nu }\left(z\right)$ defined as in §10.2(ii), ►

$${𝒞}_{\nu -1}\left(z\right)+{𝒞}_{\nu +1}\left(z\right)=(2\nu /z){𝒞}_{\nu }\left(z\right),$$

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$${𝒞}_{\nu -1}\left(z\right)-{𝒞}_{\nu +1}\left(z\right)=2{𝒞}_{\nu }^{\prime }\left(z\right).$$

►

$${𝒞}_{\nu }^{\prime }\left(z\right)={𝒞}_{\nu -1}\left(z\right)-(\nu /z){𝒞}_{\nu }\left(z\right),$$

… ►For results on modified quotients of the form $z{𝒞}_{\nu \pm 1}\left(z\right)/{𝒞}_{\nu }\left(z\right)$ see Onoe (1955) and Onoe (1956). …

8: 12.6 Continued Fraction

§12.6 Continued Fraction

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9: 12 Parabolic Cylinder Functions

Chapter 12 Parabolic Cylinder Functions

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10: 12.18 Methods of Computation

§12.18 Methods of Computation

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