31 Heun Functions Properties 31.11 Expansions in Series of Hypergeometric Functions 31.13 Asymptotic Approximations

§31.12 Confluent Forms of Heun’s Equation

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Keywords:: Heun’s equation, confluent forms
Notes:: The process of confluence is discussed in Ince (1926). See Decarreau et al. (1978a, b) for the classification of confluent forms.
Referenced by:: §31.10(ii)
Permalink:: http://dlmf.nist.gov/31.12
See also:: Annotations for Ch.31

Confluent forms of Heun’s differential equation (31.2.1) arise when two or more of the regular singularities merge to form an irregular singularity. This is analogous to the derivation of the confluent hypergeometric equation from the hypergeometric equation in §13.2(i). There are four standard forms, as follows:

Confluent Heun Equation

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Keywords:: Coulomb spheroidal functions, Heun’s equation, Mathieu functions, as confluent Heun functions, confluent Heun equation, confluent Heun functions, confluent forms, relations to other functions, special cases, spheroidal wave functions
See also:: Annotations for §31.12 and Ch.31

31.12.1		$\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\frac{\gamma}{z}+\frac{% \delta}{z-1}+\epsilon\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{\alpha z-q}{% z(z-1)}w=0.$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $q$ : real or complex parameter and $\alpha$ : real or complex parameter Referenced by: §31.12 Permalink: http://dlmf.nist.gov/31.12.E1 Encodings: TeX, pMML, png See also: Annotations for §31.12, §31.12 and Ch.31

This has regular singularities at $z=0$ and $1$ , and an irregular singularity of rank 1 at $z=\infty$ .

Mathieu functions (Chapter 28), spheroidal wave functions (Chapter 30), and Coulomb spheroidal functions (§30.12) are special cases of solutions of the confluent Heun equation.

Doubly-Confluent Heun Equation

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Keywords:: Heun’s equation, doubly-confluent, doubly-confluent Heun equation
See also:: Annotations for §31.12 and Ch.31

31.12.2		$\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\frac{\delta}{z^{2}}+\frac{% \gamma}{z}+1\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{\alpha z-q}{z^{2}}w=0.$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $q$ : real or complex parameter and $\alpha$ : real or complex parameter Permalink: http://dlmf.nist.gov/31.12.E2 Encodings: TeX, pMML, png See also: Annotations for §31.12, §31.12 and Ch.31

This has irregular singularities at $z=0$ and $\infty$ , each of rank $1$ .

Biconfluent Heun Equation

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Keywords:: Heun’s equation, biconfluent, biconfluent Heun equation
See also:: Annotations for §31.12 and Ch.31

31.12.3		$\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-\left(\frac{\gamma}{z}+\delta+z% \right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{\alpha z-q}{z}w=0.$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $q$ : real or complex parameter and $\alpha$ : real or complex parameter Referenced by: Erratum (V1.0.7) for Equation (31.12.3) Permalink: http://dlmf.nist.gov/31.12.E3 Encodings: TeX, pMML, png Errata (effective with 1.0.7): Originally the sign in front of the second term in this equation was $+$ . The correct sign is $-$ . Reported 2013-10-31 by Henryk Witek See also: Annotations for §31.12, §31.12 and Ch.31

This has a regular singularity at $z=0$ , and an irregular singularity at $\infty$ of rank $2$ .

Triconfluent Heun Equation

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Keywords:: Heun’s equation, applications, confluent Heun equation, confluent forms, properties of solutions, triconfluent, triconfluent Heun equation
See also:: Annotations for §31.12 and Ch.31

31.12.4		$\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\gamma+z\right)z\frac{% \mathrm{d}w}{\mathrm{d}z}+\left(\alpha z-q\right)w=0.$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $\gamma$ : real or complex parameter, $q$ : real or complex parameter and $\alpha$ : real or complex parameter Referenced by: §31.12 Permalink: http://dlmf.nist.gov/31.12.E4 Encodings: TeX, pMML, png See also: Annotations for §31.12, §31.12 and Ch.31

This has one singularity, an irregular singularity of rank $3$ at $z=\infty$ .

For properties of the solutions of (31.12.1)–(31.12.4), including connection formulas, see Bühring (1994), Ronveaux (1995, Parts B,C,D,E), Wolf (1998), Lay and Slavyanov (1998), and Slavyanov and Lay (2000).

31.11 Expansions in Series of Hypergeometric Functions 31.13 Asymptotic Approximations

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§31.12 Confluent Forms of Heun’s Equation

Confluent Heun Equation

Doubly-Confluent Heun Equation

Biconfluent Heun Equation

Triconfluent Heun Equation

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