§19.18 Derivatives and Differential Equations

§19.18(ii) Differential Equations

More concisely, if , then each of (19.16.14)–(19.16.18) and (19.16.20)–(19.16.23) satisfies Euler’s homogeneity relation:

and also a system of Euler–Poisson differential equations (of which only are independent):

or equivalently,

Here and . For group-theoretical aspects of this system see Carlson (1963, §VI). If , then elimination of between (19.18.11) and (19.18.12), followed by the substitution , produces the Gauss hypergeometric equation (15.10.1).

The next four differential equations apply to the complete case of and in the form (see (19.16.20) and (19.16.23)).

The function satisfies an Euler–Poisson–Darboux equation:

Also , with , satisfies a wave equation:

Similarly, the function satisfies an equation of axially symmetric potential theory:

and , with , satisfies Laplace’s equation:

19.18.17