---
tag: A1H48DDW49
kind: concept
name: Conjugate function
msc: [42A50, 31A05]
sources:
  - url: https://en.wikipedia.org/wiki/Harmonic_conjugate
---

## Summary

Let $f$ be an integrable function on the circle, and extend it to a harmonic function in the unit disc.
The conjugate function $\tilde f$ is the boundary value of the [[harmonic conjugate|050F2FMTKF]] of this extension.
On Fourier series it acts by $e^{ik\theta} \mapsto -i \sgn(k) e^{ik\theta}$, so it turns $\cos k\theta$ into $\sin k\theta$.
It is the circle version of the [[Hilbert transform|NSCEBXM7ZM]].
Its $L^p$ bounds control the convergence of [[Fourier series|G12KXDHPPS]] in $L^p$.

## Explanations

### L1 human

Every real function $f$ on the circle is the real part of the boundary values of some holomorphic function in the disc.
The imaginary part of these boundary values is the conjugate function $\tilde f$.
In terms of waves, it shifts every cosine wave into the matching sine wave, and every sine wave into minus the cosine wave.
For example, the holomorphic function $z = re^{i\theta}$ equals $r\cos\theta + i r\sin\theta$.
On the circle $r = 1$ its real part is $\cos\theta$ and its imaginary part is $\sin\theta$, so the conjugate of $\cos\theta$ is $\sin\theta$.