Complex Analysis Question

by tylerc1991
Tags: analysis, complex
 P: 1 I think we should be a little more careful with our labels, say we have a complex number $$z = re^{i\theta}$$, then $$z^2 = r^2e^{i2\theta}$$, but $$z^2 + 1 = \zeta$$ will be a new complex number we are examining with a different radius $$R$$ and phase $$\phi$$, we can begin with a form $$\zeta = u(x,y) + iv(x,y)$$, but the form of $$g(z)$$ is in terms of polar quantities, so it would probably be best to go back to a form $$\zeta = Re^{i\phi}$$ so you may directly insert those expressions for $$R$$, $$\phi$$ directly into $$g(z) \rightarrow g(\zeta ) = g(\R,\phi )$$ for $$r$$ and "$$\theta$$. Note that the new parameters $$R = R(r)$$ and $$\phi = \phi (r,\theta )$$. This approach should work I think, where the proof may be furnished by recalling the results for part (a) which proves the analyticity of $$z$$ itself (i.e. $$r$$ and $$\theta$$ ). I have not thought it through that much, but it sounds ok to me so far.