Q1Advanced Engineering Mathematics II

Question

Prove that the function $u = x^{3} - 3 x y^{2} + 3 x^{2} - 3 y^{2} + 1$ satisfies Laplace equation and determine the corresponding analytic function $f (z)$ .

Answer 1

Question

4 marks

Prove that the function $u = x^{3} - 3 x y^{2} + 3 x^{2} - 3 y^{2} + 1$ satisfies Laplace equation and determine the corresponding analytic function $f (z)$ .

Answer

u = x^3 - 3xy^2 + 3x^2 - 3y^2 + 1 satisfies Laplace's equation. The corresponding analytic function is f(z) = z^3 + 3z^2 + 1 + iC.