---
tag: W6E15N91Q3
kind: definition
name: Kummer's equation
msc: [33C15, 34M03]
defines: NT917KFR57
uses: [0QQRP17Z7S, H96HXG6CA0, AT1W4MMTYR]
sources:
  - url: https://en.wikipedia.org/wiki/Confluent_hypergeometric_function
---

## Definition

Let $a$ and $b$ be complex numbers.
*Kummer's equation* with parameters $a$ and $b$ is the [[linear differential equation|0QQRP17Z7S]] of second order
$$z \frac{d^2 w}{dz^2} + (b - z) \frac{dw}{dz} - a w = 0$$
for a function $w$ of a complex variable $z$.

## Remarks

The point $z = 0$ is a [[regular singular point|H96HXG6CA0]] with exponents $0$ and $1 - b$.
The point $z = \infty$ is an [[irregular singularity|AT1W4MMTYR]], 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.