Function Repository Resource:

# HarmonicConjugate (1.0.0)current version: 2.0.0 »

Compute the harmonic conjugate of a function

Contributed by: Jordan Hasler, Wolfram|Alpha Math Team
 ResourceFunction["HarmonicConjugate"][f,{x,y}] returns a harmonic conjugate of the function f with respect to x and y.

## Details

A harmonic function f is a function that satisfies the Laplace equation.
A function u(x,y) and a harmonic conjugate v(x,y) satisfy the Cauchy-Riemann equations, xu=yv and yu=-xv, signifying that f(x,y)=u(x,y)+iv(x,y) is a differentiable function in the complex plane.
The function returns the harmonic conjugate with constant 0.

## Examples

### Basic Examples (1)

Compute a harmonic conjugate of a polynomial function:

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### Scope (2)

Compute a harmonic conjugate of a rational function:

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Visualize the harmonic conjugate:

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### Possible Issues (1)

The function returns unevaluated if the function is not harmonic:

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## Publisher

Wolfram|Alpha Math Team

## Version History

• 2.0.0 – 23 March 2023
• 1.0.0 – 09 December 2022

## Author Notes

To view the full source code for HarmonicConjugate, evaluate the following:

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