Question

Show that f′(z) dne at any point z if (a) f(z)=z−iz⁻, (b) f(z)=Im z, (a) f(z)=eˣe⁻ᶦʸ

Answer 1

To show that f'(z) does not exist at any point z, we need to determine the partial derivatives of f(z) with respect to x and y and check if they satisfy the Cauchy-Riemann equations.

Step-by-step explanation:

To show that f'(z) does not exist at any point z, we need to determine the partial derivatives of f(z) with respect to x and y and check if they satisfy the Cauchy-Riemann equations.

(a) For f(z) = z - iz^(-1), the partial derivatives are:

∂f/∂x = 1, ∂f/∂y = -i/z^2

These partial derivatives do not satisfy the Cauchy-Riemann equations, hence f'(z) does not exist at any point z.