Q1Advanced Engineering Mathematics II

Question

Prove that the function $f (z) = u + i v$ ; where $f (z) = \frac{x ^{2} y ^{5} ( x + i y )}{x ^{4} + y ^{10}}, z \neq = 0$ $f (0) = 0$ satisfy Cauchy- Riemann equations at origin, but $f^{'} (0)$ does not exist.

Answer 1

Question

10 marks

Prove that the function $f (z) = u + i v$ ; where $f (z) = \frac{x ^{2} y ^{5} ( x + i y )}{x ^{4} + y ^{10}}, z \neq = 0$ $f (0) = 0$ satisfy Cauchy- Riemann equations at origin, but $f^{'} (0)$ does not exist.

Answer

The function f(z) = x^2y^5(x+iy)/(x^4+y^10) satisfies C-R equations at origin by direct computation, but f'(0) does not exist because the limit depends on the path of approach.