---
tag: NT917KFR57
kind: concept
name: Kummer's equation
msc: [33C15, 34M03]
specializes: [0QQRP17Z7S]
sources:
  - url: https://en.wikipedia.org/wiki/Confluent_hypergeometric_function
---

## Summary

Kummer's equation is the second order linear differential equation $z w'' + (b - z) w' - a w = 0$, with complex parameters $a$ and $b$.
It has a [[regular singular point|H96HXG6CA0]] at $0$ and an [[irregular singularity|AT1W4MMTYR]] at $\infty$.
Its solutions are the confluent hypergeometric functions, which include Laguerre polynomials, Hermite functions, the error function and the incomplete gamma function.

## Explanations

### L1 human

Kummer's equation arises when two of the three singular points of the hypergeometric equation are pushed together.
One replaces $z$ by $z / \beta$ in the hypergeometric equation with parameters $a, \beta, b$, and lets $\beta \to \infty$; the singular points $\beta$ and $\infty$ merge into one.
The word "confluent" refers to this merging.
A small example: for $a = b$, the function $w = e^z$ solves the equation, since $z e^z + (b - z) e^z - b e^z = 0$.