---
tag: N3CC5DRGT1
kind: concept
name: Hankel functions
msc: [33C10, 34B30]
sources:
  - url: https://en.wikipedia.org/wiki/Bessel_function
---

## Summary

The Hankel functions $H^{(1)}_\nu$ and $H^{(2)}_\nu$ are the Bessel functions of the third kind.
They are the complex combinations $J_\nu \pm i Y_\nu$ of the [[Bessel functions of the first kind|CZCMRDWDYW]] and of the [[second kind|GRSNDJG363]].
They play the role of $e^{\pm i x}$ next to the roles of $\cos x$ and $\sin x$ played by $J_\nu$ and $Y_\nu$.
They describe outgoing and incoming cylindrical waves.

## Explanations

### L1 human

For the equation $y'' + y = 0$, one can use the real solutions $\cos x$ and $\sin x$ or the complex solutions $e^{ix}$ and $e^{-ix}$.
Bessel's equation has the same choice.
The pair $J_\nu$, $Y_\nu$ is the real one, and the Hankel functions are the complex one.
For large $x > 0$, $H^{(1)}_\nu(x)$ is close to $\sqrt{\frac{2}{\pi x}}\, e^{i(x - \nu\pi/2 - \pi/4)}$, a wave travelling outward whose size shrinks.