Question

a) Show using the definition that f(z) = z² is analytic and compute itsderivative. b) Which of the following functions are complex differentiable on the whole of the complex plane? (You must justify your answers for credit.) ) Suppose that an analytic function f defined on the whole of C satisfies Im(f(z)) = 0 for all z E C. Show that f is a constant function. [7 If a complex function f(z) can be written in the form f(x+i y)=u(x, y)+i v(x, y) \text { and if } u(x, y)=x^{3}-3 x y^{2}-x \text {, show that } u(x, y) \text { is a } harmonic function and find its harmonic conjugate, v(x, y). f(z) = Re(z)ii) f(z) = ziii)/(z) = Re (2²)

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