Construct an analytic function whose real part is u x3 3xy

Construct an analytic function whose real part is u = x^3 - 3xy^2 - 5y.

Solution

let the required analytic function be f(z) = u + iv

Given u = x³ - 3xy² -5y

Since f is analyticby Cauchy - Riemann equations u_x = v_y and u_y = -v_x

Here u_x = 3x² - 3y²= v_{y.......................(1)}

u_y = -6xy -5 = -v_{x......................(2)}

_{Integrating equation (1) with respect to y}

v = 3x²y - y³

Integrating equation (2) with respect to x

v= 3xy² + 5y

So one analytic function with the given real part is f(z) =  x³ - 3xy² -5y + i(3x²y - y³)

Another is f(z) =  x³ - 3xy² -5y + i(3xy² + 5y)