Kummer's equation

Remarks

The point $z = 0$ is a regular singular point with exponents $0$ and $1 - b$. The point $z = \infty$ is an irregular singularity, and there are no other singular points. Example: for $a = b$, the function $w = e^z$ is a solution, since $z e^z + (b - z) e^z - b e^z = 0$. Non-example: for $a \ne b$, the same function is not a solution, since the left side equals $(b - a) e^z \ne 0$. For $b$ not in $\set{0, -1, -2, \dots}$, the solution analytic at $0$ with value $1$ is Kummer's function $M(a, b, z) = {}_1F_1(a; b; z)$. A second standard solution is the Tricomi function $U(a, b, z)$, singled out by its behaviour $U(a, b, z) \sim z^{-a}$ as $z \to \infty$ in the right half-plane. The substitution $w = z^{-b/2} e^{z/2} v$ turns the equation into Whittaker's equation, which has no first derivative term.