Heun equation

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11: 31.16 Mathematical Applications

§31.16 Mathematical Applications

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§31.16(i) Uniformization Problem for Heun’s Equation

… ►It describes the monodromy group of Heun’s equation for specific values of the accessory parameter. … ►

31.16.3 $$A_0=\frac{n!}{{\left(\gamma +\delta \right)}_n}{\mathrm{𝐻𝑝}}_{n,m}\left(1\right),Q_0{A}_0+R_0{A}_1=0,$$

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Symbols:

${\mathrm{𝐻𝑝}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$: Heun polynomials, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $m$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter, $\beta$: real or complex parameter, $A_j$: coefficients, $Q_j$ and $R_j$

Referenced by:

§31.16(ii), Erratum (V1.0.21) for Equations (31.16.2) and (31.16.3), Erratum (V1.0.21) for Equations (31.16.2) and (31.16.3)

Permalink:

http://dlmf.nist.gov/31.16.E3

Encodings:

pMML, png, TeX

Correction (effective with 1.0.21):

The initial data $A_0$, previously missing, has now been included.

See also:

Annotations for §31.16(ii), §31.16 and Ch.31

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12: 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 …

13: 31.8 Solutions via Quadratures

§31.8 Solutions via Quadratures

… ►Here ${\mathrm{\Psi }}_{g,N}(\lambda ,z)$ is a polynomial of degree $g$ in $\lambda$ and of degree $N=m_0+m_1+m_2+m_3$ in $z$, that is a solution of the third-order differential equation satisfied by a product of any two solutions of Heun’s equation. …Lastly, ${\lambda }_j$, $j=1,2,\mathrm{\dots },2g+1$, are the zeros of the Wronskian of $w_{+}(𝐦;\lambda ;z)$ and $w_{-}(𝐦;\lambda ;z)$. … ►For $𝐦=(m_0,0,0,0)$, these solutions reduce to Hermite’s solutions (Whittaker and Watson (1927, §23.7)) of the Lamé equation in its algebraic form. …For more details see Smirnov (2002). …

14: 31.4 Solutions Analytic at Two Singularities: Heun Functions

… ►The eigenvalues $q_m$ satisfy the continued-fraction equation …

15: 31.11 Expansions in Series of Hypergeometric Functions

§31.11 Expansions in Series of Hypergeometric Functions

… ►Let $w(z)$ be any Fuchs–Frobenius solution of Heun’s equation. … ►Every Fuchs–Frobenius solution of Heun’s equation (31.2.1) can be represented by a series of Type I. … ►

§31.11(v) Doubly-Infinite Series

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16: 31.3 Basic Solutions

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

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

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31.3.10 $$z^{-\alpha }H\mathrm{\ell }(\frac{1}{a},\frac{q}{a}-\alpha \left(\beta -ϵ\right)-\frac{\alpha }{a}\left(\beta -\delta \right);\alpha ,\alpha -\gamma +1,\alpha -\beta +1,\delta ;\frac{1}{z}),$$

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Symbols:

$H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Referenced by:

§31.11(iii), §31.11(iii), Erratum (V1.1.7) for Equations (31.3.10), (31.3.11), Erratum (V1.1.7) for Subsection 31.11(iii)

Permalink:

http://dlmf.nist.gov/31.3.E10

Encodings:

pMML, png, TeX

Correction (effective with 1.1.7):

The second entry in the $H\mathrm{\ell }$ has been corrected with an extra minus sign.

See also:

Annotations for §31.3(ii), §31.3 and Ch.31

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31.3.11 $$z^{-\beta }H\mathrm{\ell }(\frac{1}{a},\frac{q}{a}-\beta \left(\alpha -ϵ\right)-\frac{\beta }{a}\left(\alpha -\delta \right);\beta ,\beta -\gamma +1,\beta -\alpha +1,\delta ;\frac{1}{z}).$$

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Symbols:

$H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Referenced by:

§31.3(iii), Erratum (V1.1.7) for Equations (31.3.10), (31.3.11)

Permalink:

http://dlmf.nist.gov/31.3.E11

Encodings:

pMML, png, TeX

Correction (effective with 1.1.7):

The second entry in the $H\mathrm{\ell }$ has been corrected with an extra minus sign.

See also:

Annotations for §31.3(ii), §31.3 and Ch.31

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§31.3(iii) Equivalent Expressions

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17: Bibliography E

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Integral equations for Heun functions. Quart. J. Math., Oxford Ser. 13, pp. 107–112.

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External Links:

Document, MathReview (E. Rothe), ZentralBlatt

Referenced by:

§31.10(i).

Encodings:

BibTeX

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Certain expansions of solutions of the Heun equation. Quart. J. Math., Oxford Ser. 15, pp. 62–69.

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External Links:

Document, MathReview (R. E. Langer), ZentralBlatt

Referenced by:

§31.11(i), §31.11(i), §31.9(i).

Encodings:

BibTeX

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18: 29.2 Differential Equations

… ►For the Weierstrass function $\mathrm{\wp }$ see §23.2(ii). ►Equation (29.2.10) is a special case of Heun’s equation (31.2.1). …

19: Bibliography T

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Hyperspherical elliptic harmonics and their relation to the Heun equation. Phys. Rev. A 63 (032510), pp. 1–8.

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External Links:

Document

Referenced by:

§31.17(ii).

Encodings:

BibTeX

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20: Bibliography F

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Asymptotics of the spectrum of the Heun equation and of Heun functions. Izv. Akad. Nauk SSSR Ser. Mat. 55 (3), pp. 631–646 (Russian).

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Notes:

In Russian. English translation: Math. USSR-Izv., 38(1992), no. 3, pp. 621–635 (1992)

External Links:

ISSN 0373-2436, Document, MathReview (Andreĭ Bolibrukh), ZentralBlatt

Referenced by:

§31.13.

Encodings:

BibTeX

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