Heun functions and Heun equation

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11: Brian D. Sleeman

… ► thesis was Some Boundary Value Problems Associated with the Heun Equation. … ► Plank) of Differential equations and mathematical biology, published by CRC Press in 2003, with a second edition in 2010. … ►

31 Heun Functions

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12: 31.16 Mathematical Applications

§31.16 Mathematical Applications

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

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13: 31.14 General Fuchsian Equation

§31.14 General Fuchsian Equation

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§31.14(i) Definitions

… ►Heun’s equation (31.2.1) corresponds to $N=3$. ►

Normal Form

… ►The algorithm returns a list of solutions if they exist. …

14: 31.8 Solutions via Quadratures

… ►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. …

15: 31.11 Expansions in Series of Hypergeometric Functions

§31.11 Expansions in Series of Hypergeometric Functions

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$\mu =\gamma +\delta -2.$

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§31.11(v) Doubly-Infinite Series

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16: 31.2 Differential Equations

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§31.2(i) Heun’s Equation

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31.2.1 $$\frac{d^{2}w}{{dz}^2}+\left(\frac{\gamma }{z}+\frac{\delta }{z-1}+\frac{ϵ}{z-a}\right)\frac{dw}{dz}+\frac{\alpha \beta z-q}{z(z-1)(z-a)}w=0,$$ $\alpha +\beta +1=\gamma +\delta +ϵ$.

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: 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:

§29.2(ii), §31.11(iii), §31.12, §31.13, §31.14(i), §31.17(i), §31.18, §31.2(v), §31.2(v), §31.2(v), §31.2(i), §31.3(i), §31.3(i), §31.3(ii), §31.3(ii), §31.3(ii), §31.3(iii), §31.3(iii), §31.4, §31.5, §31.6, §31.7(ii), §31.8, §31.8, §31.8

Permalink:

http://dlmf.nist.gov/31.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.2(i), §31.2 and Ch.31

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§31.2(ii) Normal Form of Heun’s Equation

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§31.2(v) Heun’s Equation Automorphisms

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Composite Transformations

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17: 31.3 Basic Solutions

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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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18: 31.7 Relations to Other Functions

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§31.7(i) Reductions to the Gauss Hypergeometric Function

… ►Other reductions of $H\mathrm{\ell }$ to a ${}_2{}^{}F_1^{}$, with at least one free parameter, exist iff the pair $(a,p)$ takes one of a finite number of values, where $q=\alpha \beta p$. … ►

§31.7(ii) Relations to Lamé Functions

… ►equation (31.2.1) becomes Lamé’s equation with independent variable $\zeta$; compare (29.2.1) and (31.2.8). The solutions (31.3.1) and (31.3.5) transform into even and odd solutions of Lamé’s equation, respectively. …

19: 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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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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