Function Repository Resource:

HarmonicConjugate

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:

In[1]:=

$ResourceFunction["HarmonicConjugate"][x^2 - y^2, {x, y}]$

Out[1]=

Scope (2)

Compute a harmonic conjugate of a rational function:

In[2]:=

$ResourceFunction["HarmonicConjugate"][-x/(x^2 + y^2), {x, y}]$

Out[2]=

Visualize the harmonic conjugate:

In[3]:=

$ContourPlot[y/(x^2 + y^2), {x, -2, 2}, {y, -2, 2}]$

Out[3]=

Possible Issues (1)

The function returns unevaluated if the function is not harmonic:

In[4]:=

$ResourceFunction["HarmonicConjugate"][x^2 + y^2, {x, y}]$

Out[4]=

Publisher

Wolfram|Alpha Math Team

Version History

2.0.0 – 23 March 2023
1.0.0 – 09 December 2022

Related Resources

Author Notes

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