biconfluent%20Heun%20equation

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11: 31.17 Physical Applications

§31.17 Physical Applications

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§31.17(i) Addition of Three Quantum Spins

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§31.17(ii) Other Applications

… ►For application of biconfluent Heun functions in a model of an equatorially trapped Rossby wave in a shear flow in the ocean or atmosphere see Boyd and Natarov (1998).

12: 20 Theta Functions

Chapter 20 Theta Functions

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13: 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).

14: 28 Mathieu Functions and Hill’s Equation

Chapter 28 Mathieu Functions and Hill’s Equation

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15: 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. …

16: 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 $$(s_1,s_2){\mathrm{𝐻𝑓}}_m^{\nu }(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z),$$ $m=0,1,2,\mathrm{\dots }$,

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Symbols:

$(s_1,s_2){\mathrm{𝐻𝑓}}_m(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z)$: 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

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http://dlmf.nist.gov/31.6.E1

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See also:

Annotations for §31.6 and Ch.31

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17: 8 Incomplete Gamma and Related
Functions

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18: 8.26 Tables

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

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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 $(x+n){\mathrm{e}}^x{E}_n\left(x\right)$ 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; ${\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2(1)10$, $x=10(.1)20$ to 7D; ${\mathrm{e}}^x{E}_p\left(x\right)$ for $p=0(.1)1$, $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

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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.

19: 23 Weierstrass Elliptic and Modular
Functions

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20: 31.3 Basic Solutions

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

► $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$ denotes the solution of (31.2.1) that corresponds to the exponent $0$ at $z=0$ and assumes the value $1$ there. … ►

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

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§31.3(iii) Equivalent Expressions

… ►For example, $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$ is equal to …