§31.12 Confluent Forms of Heun’s Equation

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

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

Keywords:: Coulomb spheroidal functions, Heun’s equation, Mathieu functions, confluent Heun equation, spheroidal wave functions

31.12.1 $\frac{{d}^{2}w}{{dz}^{2}}+\left(\frac{\gamma}{z}+\frac{\delta}{z-1}+\epsilon\right)\frac{dw}{dz}+\frac{\alpha z-q}{z(z-1)}w=0.$

Symbols:: $\frac{df}{dx}$ : 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

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

Keywords:: Heun’s equation, doubly-confluent Heun equation

31.12.2 $\frac{{d}^{2}w}{{dz}^{2}}+\left(\frac{\delta}{z^{2}}+\frac{\gamma}{z}+1\right)\frac{dw}{dz}+\frac{\alpha z-q}{z^{2}}w=0.$

Symbols:: $\frac{df}{dx}$ : 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

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

¶ Biconfluent Heun Equation

Keywords:: Heun’s equation, biconfluent Heun equation

31.12.3 $\frac{{d}^{2}w}{{dz}^{2}}+\left(\frac{\gamma}{z}+\delta+z\right)\frac{dw}{dz}+\frac{\alpha z-q}{z}w=0.$

Symbols:: $\frac{df}{dx}$ : 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.E3
Encodings:: TeX, pMML, png

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

¶ Triconfluent Heun Equation

Keywords:: Heun’s equation, confluent Heun equation, triconfluent Heun equation

31.12.4 $\frac{{d}^{2}w}{{dz}^{2}}+\left(\gamma+z\right)z\frac{dw}{dz}+\left(\alpha z-q\right)w=0.$

Symbols:: $\frac{df}{dx}$ : 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

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).