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Harmonic function and conjugate

Short answer

A function is harmonic if it satisfies Laplace's equation ∂²u/∂x²+∂²u/∂y²=0. For u = eˣ cos y, ∂²u/∂x² = eˣ cos y, ∂²u/∂y² = −eˣ cos y. Their sum is 0, so u is harmonic.

Define harmonic function. Show that the function u(x, y) = eˣ cos y is harmonic.

Definition

A real function u(x,y) with continuous second partial derivatives is harmonic in a region if it satisfies Laplace's equation there:

∂²u/∂x² + ∂²u/∂y² = 0.

(The real and imaginary parts of any analytic function are harmonic.)

u = eˣ cos y

∂u/∂x = eˣ cos y      ∂²u/∂x² = eˣ cos y
∂u/∂y = −eˣ sin y     ∂²u/∂y² = −eˣ cos y

∂²u/∂x² + ∂²u/∂y² = eˣ cos y − eˣ cos y = 0

So u = eˣ cos y is harmonic. It is the real part of the analytic function f(z) = eᶻ = eˣ(cos y + i sin y).

✓ Answer checked by Claude Opus 5.5 (high effort)

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