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31 Heun FunctionsProperties

§31.12 Confluent Forms of Heun’s Equation

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

31.12.1 2wz2+(γz+δz-1+ϵ)wz+αz-qz(z-1)w=0.

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

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

31.12.2 2wz2+(δz2+γz+1)wz+αz-qz2w=0.

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

Biconfluent Heun Equation

31.12.3 2wz2-(γz+δ+z)wz+αz-qzw=0.

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

Triconfluent Heun Equation

31.12.4 2wz2+(γ+z)zwz+(αz-q)w=0.

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

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).