---
tag: GRSNDJG363
kind: concept
name: Bessel function of the second kind
msc: [33C10, 34B30]
sources:
  - url: https://en.wikipedia.org/wiki/Bessel_function
---

## Summary

The Bessel function of the second kind $Y_\nu$ is a second solution of Bessel's equation, independent of the [[Bessel function of the first kind|CZCMRDWDYW]] $J_\nu$.
It is also called the Weber function or the Neumann function, and it is sometimes written $N_\nu$.
Unlike $J_\nu$, it is unbounded near $0$.
Together, $J_\nu$ and $Y_\nu$ span all solutions of [[Bessel's equation|EAV97QH9DF]].

## Explanations

### L1 human

Bessel's equation is of second order, so it needs two independent solutions.
The first kind $J_\nu$ is the one that stays finite at the origin.
The second kind $Y_\nu$ is the other one, and it blows up at the origin.
For example, $Y_0(x)$ behaves like $\frac{2}{\pi} \ln x$ for small $x > 0$.
For large $x$, both $J_\nu$ and $Y_\nu$ oscillate like shifted sine waves of slowly shrinking size.