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## Homework Statement

Suppose v is a harmonic conjugate of u in a domain D, and that u is a harmonic conjugate of v in D. Show how it follows that u(x,y) and v(x,y) are constant throughout D.

## The Attempt at a Solution

since u is a harmonic conjugate of v, u_xx + u_yy = 0

also, since v is a harmonic conjugate of u, v_xx + v_yy = 0

u_x = v_y => u_xx = v_yx

u_y = -v_x => u_yy = -v_xy

I think that I am going in circles here. Can someone lend a helping hand with this problem? Thank you so much!