Sharpe's Differential Equation

A generalization of the Bessel Differential Equation for functions of order 0, given by

$$zy''+y'+(z+A)y=0.$$

Solutions are

$$y=e^{\pm iz} {}_1F_1\left({{\textstyle{1\over 2}}\mp {\textstyle{1\over 2}}iA; 1; \mp 2iz}\right),$$

where ${}_1F_1(a;b;x)$ is a Confluent Hypergeometric Function.

See also Bessel Differential Equation, Confluent Hypergeometric Function