Answer in Chemical Engineering for pavani #231376
March 11th, 2023
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).
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).