# §31.2 Differential Equations

## §31.2(i) Heun’s Equation

This equation has regular singularities at , with corresponding exponents , , , , respectively (§2.7(i)). All other homogeneous linear differential equations of the second order having four regular singularities in the extended complex plane, , can be transformed into (31.2.1).

The parameters play different roles: is the singularity parameter; are exponent parameters; is the accessory parameter. The total number of free parameters is six.

31.2.5

## §31.2(iv) Doubly-Periodic Forms

### ¶ Jacobi’s Elliptic Form

With the notation of §22.2 let

Then (suppressing the parameter )

## §31.2(v) Heun’s Equation Automorphisms

### ¶ -Homotopic Transformations

satisfies (31.2.1) if is a solution of (31.2.1) with transformed parameters ; , , . Next, satisfies (31.2.1) if is a solution of (31.2.1) with transformed parameters ; , , . Lastly, satisfies (31.2.1) if is a solution of (31.2.1) with transformed parameters ; , , . By composing these three steps, there result possible transformations of the dependent variable (including the identity transformation) that preserve the form of (31.2.1).

### ¶ Homographic Transformations

There are homographies that take to some permutation of , where may differ from . If is one of the homographies that map to , then satisfies (31.2.1) if is a solution of (31.2.1) with replaced by and appropriately transformed parameters. For example, if , then the parameters are , ; , . If is one of the homographies that do not map to , then an appropriate prefactor must be included on the right-hand side. For example, , which arises from , satisfies (31.2.1) if is a solution of (31.2.1) with replaced by and transformed parameters , ; , .

### ¶ Composite Transformations

There are automorphisms of equation (31.2.1) by compositions of -homotopic and homographic transformations. Each is a substitution of dependent and/or independent variables that preserves the form of (31.2.1). Except for the identity automorphism, each alters the parameters.