Answer to Question #231376 in Chemical Engineering for pavani

Question

2) Show that v(x,y)=x²-y²-y is harmonic function.Find it's conjugate harmonic function u(x,y) and corresponding analytic function f(z).

Answer 1

Let z = x + iy and write f (z) = u(x, y) + iv(x, y).

If f (z) = u(x, y) + iv(x, y) is analytic on a region A then both u and v are

harmonic functions on A.

 Since ux = vy

vxy = vyx.

To fill the tight link between analytic and harmonic functions, we show that any harmonic function is the real part of an analytic function.

If u(x,y) is harmonic on a simply connected region A, then u is the real part of an analytic function f (z) = u(x, y) + iv(x, y).