Question
Show that $\log \left(x^2+y^2\right)$ and $\frac{x}{x^2+y^2}$ are harmonic functions, that is, solutions of the two-dimensional Laplace equation.
Step 1
We need to show that both \( u \) and \( v \) satisfy the Laplace equation, which is given by: \[ \Delta f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = 0 \] for a function \( f(x, y) \). Show more…
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