triconfluent%20Heun%20equation

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11: 20 Theta Functions

Chapter 20 Theta Functions

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12: 31.13 Asymptotic Approximations

§31.13 Asymptotic Approximations

… ►For asymptotic approximations of the solutions of Heun’s equation (31.2.1) when two singularities are close together, see Lay and Slavyanov (1999). ►For asymptotic approximations of the solutions of confluent forms of Heun’s equation in the neighborhood of irregular singularities, see Komarov et al. (1976), Ronveaux (1995, Parts B,C,D,E), Bogush and Otchik (1997), Slavyanov and Veshev (1997), and Lay et al. (1998).

13: 28 Mathieu Functions and Hill’s Equation

Chapter 28 Mathieu Functions and Hill’s Equation

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14: 31.4 Solutions Analytic at Two Singularities: Heun Functions

§31.4 Solutions Analytic at Two Singularities: Heun Functions

… ►To emphasize this property this set of functions is denoted by … ►The eigenvalues

q_{m}

satisfy the continued-fraction equation … ►The set

q_{m}

depends on the choice of

s_{1}

and

s_{2}

. ►The solutions (31.4.3) are called the Heun functions. …

15: 31.6 Path-Multiplicative Solutions

§31.6 Path-Multiplicative Solutions

►A further extension of the notation (31.4.1) and (31.4.3) is given by ►

31.6.1

(s_{1},s_{2})\mathit{Hf}_{m}^{\nu}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z% \right),

m=0,1,2,\dots

ⓘ

Symbols:: $(\NVar{s_{1}},\NVar{s_{2}})\mathit{Hf}_{\NVar{m}}\left(\NVar{a},\NVar{q_{m}};% \NVar{\alpha},\NVar{\beta},\NVar{\gamma},\NVar{\delta};\NVar{z}\right)$ : Heun functions, $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\nu$ : real or complex parameter, $m$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter, $\alpha$ : real or complex parameter, $\beta$ : real or complex parameter and $s_{j}$ : parameters
Permalink:: http://dlmf.nist.gov/31.6.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §31.6 and Ch.31

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16: 8 Incomplete Gamma and Related
Functions

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17: 8.26 Tables

… ►

•

Khamis (1965) tabulates $P\left(a,x\right)$ for $a=0.05(.05)10(.1)20(.25)70$ , $0.0001\leq x\leq 250$ to 10D.

… ►

•

Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_{n}\left(x\right)$ for $n=2,3,4,10,20$ , $x=0(.01)2$ to 7D; also $(x+n)e^{x}E_{n}\left(x\right)$ for $n=2,3,4,10,20$ , $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

… ►

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Pagurova (1961) tabulates $E_{n}\left(x\right)$ for $n=0(1)20$ , $x=0(.01)2(.1)10$ to 4-9S; $e^{x}E_{n}\left(x\right)$ for $n=2(1)10$ , $x=10(.1)20$ to 7D; $e^{x}E_{p}\left(x\right)$ for $p=0(.1)1$ , $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

… ►

•

Zhang and Jin (1996, Table 19.1) tabulates $E_{n}\left(x\right)$ for $n=1,2,3,5,10,15,20$ , $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

18: 23 Weierstrass Elliptic and Modular
Functions

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19: 31.3 Basic Solutions

… ►

§31.3(i) Fuchs–Frobenius Solutions at $z=0$

►

\mathit{H\!\ell}\left(a,q;\alpha,\beta,\gamma,\delta;z\right)

denotes the solution of (31.2.1) that corresponds to the exponent

0

z=0

and assumes the value

1

there. … ►

§31.3(ii) Fuchs–Frobenius Solutions at Other Singularities

… ►

§31.3(iii) Equivalent Expressions

… ►For example,

\mathit{H\!\ell}\left(a,q;\alpha,\beta,\gamma,\delta;z\right)

is equal to …

20: 31.17 Physical Applications

§31.17 Physical Applications

►

§31.17(i) Addition of Three Quantum Spins

… ►

§31.17(ii) Other Applications

… ►For applications of Heun’s equation and functions in astrophysics see Debosscher (1998) where different spectral problems for Heun’s equation are also considered. …