harmonic conjugate functions

Two harmonic functions $ u$ and $ v$ from an open subset $ A$ of $ \mathbb{R}\times\mathbb{R}$ to $ \mathbb{R}$ , which satisfy the Cauchy-Riemann equations

$\displaystyle u_x = v_y, \,\,\, u_y = -v_x,$ (1)

are the harmonic conjugate functions of each other.

Example. $ \sin{x}\cosh{y}$ and $ \cos{x}\sinh{y}$ are harmonic conjugates of each other.



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