Whittaker%E2%80%93Hill%20equation

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21: 10.39 Relations to Other Functions

… ►Principal values on each side of these equations correspond. … ►

10.39.7 $$I_{\nu }\left(z\right)=\frac{{(2z)}^{-\frac{1}{2}}{M}_{0,\nu }\left(2z\right)}{2^{2\nu }\mathrm{\Gamma }\left(\nu +1\right)},$$ $2\nu \ne -1,-2,-3,\mathrm{\dots }$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.47

Referenced by:

(9.6.23), (9.6.24)

Permalink:

http://dlmf.nist.gov/10.39.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.39, §10.39 and Ch.10

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10.39.8 $$K_{\nu }\left(z\right)={\left(\frac{\pi }{2z}\right)}^{\frac{1}{2}}{W}_{0,\nu }\left(2z\right).$$

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Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.48

Referenced by:

(9.6.21), (9.6.22)

Permalink:

http://dlmf.nist.gov/10.39.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §10.39, §10.39 and Ch.10

►For the functions $M$, $U$, $M_{0,\nu }$, and $W_{0,\nu }$ see §§13.2(i) and 13.14(i). …

22: 13.22 Zeros

§13.22 Zeros

►From (13.14.2) and (13.14.3) $M_{\kappa ,\mu }\left(z\right)$ has the same zeros as $M(\frac{1}{2}+\mu -\kappa ,1+2\mu ,z)$ and $W_{\kappa ,\mu }\left(z\right)$ has the same zeros as $U(\frac{1}{2}+\mu -\kappa ,1+2\mu ,z)$, hence the results given in §13.9 can be adopted. … ►For example, if $\mu (\ge 0)$ is fixed and $\kappa (>0)$ is large, then the $r$th positive zero ${\varphi }_r$ of $M_{\kappa ,\mu }\left(z\right)$ is given by …

23: 12.19 Tables

… ►

Kireyeva and Karpov (1961) includes $D_p\left(x(1+\mathrm{i})\right)$ for $\pm x=0(.1)5$, $p=0(.1)2$, and $\pm x=5(.01)10$, $p=0(.5)2$, 7D.

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Karpov and Čistova (1964) includes $D_p\left(x\right)$ for $p=-2(.1)0$, $\pm x=0(.01)5$; $p=-2(.05)0$, $\pm x=5(.01)10$, 6D.

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Karpov and Čistova (1968) includes ${\mathrm{e}}^{-\frac{1}{4}{x}^2}{D}_p\left(-x\right)$ and ${\mathrm{e}}^{-\frac{1}{4}{x}^2}{D}_p\left(\mathrm{i}x\right)$ for $x=0(.01)5$ and $x^{-1}$ = 0(.001 or .0001)5, $p=-1(.1)1$, 7D or 8S.

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Murzewski and Sowa (1972) includes $D_{-n}\left(x\right)$ $\left(=U(n-\frac{1}{2},x)\right)$ for $n=1(1)20$, $x=0(.05)3$, 7S.

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24: 33.14 Definitions and Basic Properties

… ►

§33.14(i) Coulomb Wave Equation

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§33.14(ii) Regular Solution $f(ϵ,\mathrm{\ell };r)$

… ►where $M_{\kappa ,\mu }\left(z\right)$ and $M(a,b,z)$ are defined in §§13.14(i) and 13.2(i), and … ►

§33.14(iii) Irregular Solution $h(ϵ,\mathrm{\ell };r)$

… ►

33.14.7 $$h(ϵ,\mathrm{\ell };r)=\frac{\mathrm{\Gamma }\left(\mathrm{\ell }+1-\kappa \right)}{\pi {\kappa }^{\mathrm{\ell }}}\left(W_{\kappa ,\mathrm{\ell }+\frac{1}{2}}\left(2r/\kappa \right)+{(-1)}^{\mathrm{\ell }}S(ϵ,r)\frac{\mathrm{\Gamma }\left(\mathrm{\ell }+1+\kappa \right)}{2(2\mathrm{\ell }+1)!}{M}_{\kappa ,\mathrm{\ell }+\frac{1}{2}}\left(2r/\kappa \right)\right),$$

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Defines:

$h(ϵ,\mathrm{\ell };r)$: irregular Coulomb function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter, $\kappa$: quantity and $S(ϵ,r)$: function

Referenced by:

§13.18(vi)

Permalink:

http://dlmf.nist.gov/33.14.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §33.14(iii), §33.14 and Ch.33

…

25: 28.8 Asymptotic Expansions for Large $q$

… ►Also let $\xi =2\sqrt{h}\cos x$ and $D_m\left(\xi \right)={\mathrm{e}}^{-{\xi }^2/4}{\mathrm{𝐻𝑒}}_m\left(\xi \right)$ (§18.3). … ►Barrett (1981) supplies asymptotic approximations for numerically satisfactory pairs of solutions of both Mathieu’s equation (28.2.1) and the modified Mathieu equation (28.20.1). … ►The approximations are expressed in terms of Whittaker functions $W_{\kappa ,\mu }\left(z\right)$ and $M_{\kappa ,\mu }\left(z\right)$ with $\mu =\frac{1}{4}$; compare §2.8(vi). …With additional restrictions on $z$, uniform asymptotic approximations for solutions of (28.2.1) and (28.20.1) are also obtained in terms of elementary functions by re-expansions of the Whittaker functions; compare §2.8(ii). ►Subsequently the asymptotic solutions involving either elementary or Whittaker functions are identified in terms of the Floquet solutions ${\mathrm{me}}_{\nu }(z,q)$ (§28.12(ii)) and modified Mathieu functions ${\mathrm{M}}_{\nu }^{(j)}(z,h)$ (§28.20(iii)). …

26: 28.32 Mathematical Applications

… ►The two-dimensional wave equation … ► … ►

§28.32(ii) Paraboloidal Coordinates

… ►is separated in this system, each of the separated equations can be reduced to the Whittaker–Hill equation (28.31.1), in which $A,B$ are separation constants. …

27: 13.32 Software

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28: 13.29 Methods of Computation

… ►Similarly for the Whittaker functions. ►

§13.29(ii) Differential Equations

… ►For $M(a,b,z)$ and $M_{\kappa ,\mu }\left(z\right)$ this means that in the sector $|\mathrm{ph}z|\le \pi$ we may integrate along outward rays from the origin with initial values obtained from (13.2.2) and (13.14.2). ►For $U(a,b,z)$ and $W_{\kappa ,\mu }\left(z\right)$ we may integrate along outward rays from the origin in the sectors , with initial values obtained from connection formulas in §13.2(vii), §13.14(vii). … ►The integral representations (13.4.1) and (13.4.4) can be used to compute the Kummer functions, and (13.16.1) and (13.16.5) for the Whittaker functions. …

29: 13.17 Continued Fractions

§13.17 Continued Fractions

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13.17.1 $$\frac{\sqrt{z}{M}_{\kappa ,\mu }\left(z\right)}{M_{\kappa -\frac{1}{2},\mu +\frac{1}{2}}\left(z\right)}=1+\frac{u_{1}z}{1+}\frac{u_{2}z}{1+}\mathrm{\cdots },$$

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Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $z$: complex variable and $u_n$: continued fraction coefficients

Permalink:

http://dlmf.nist.gov/13.17.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.17 and Ch.13

… ►

13.17.3 $$\frac{W_{\kappa ,\mu }\left(z\right)}{\sqrt{z}{W}_{\kappa -\frac{1}{2},\mu -\frac{1}{2}}\left(z\right)}=1+\frac{v_1/z}{1+}\frac{v_2/z}{1+}\mathrm{\cdots },$$

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Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $z$: complex variable and $v_n$: continued fraction coefficients

Permalink:

http://dlmf.nist.gov/13.17.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.17 and Ch.13

…

30: 12.1 Special Notation

… ►An older notation, due to Whittaker (1902), for $U(a,z)$ is $D_{\nu }\left(z\right)$. The notations are related by $U(a,z)=D_{-a-\frac{1}{2}}\left(z\right)$. Whittaker’s notation $D_{\nu }\left(z\right)$ is useful when $\nu$ is a nonnegative integer (Hermite polynomial case).