Complex Methods | Part IB, 2002

(a) Show that if ff is an analytic function at z0z_{0} and f(z0)0f^{\prime}\left(z_{0}\right) \neq 0, then ff is conformal at z0z_{0}, i.e. it preserves angles between paths passing through z0z_{0}.

(b) Let DD be the disc given by z+i<2|z+i|<\sqrt{2}, and let HH be the half-plane given by y>0y>0, where z=x+iyz=x+i y. Construct a map of the domain DHD \cap H onto HH, and hence find a conformal mapping of DHD \cap H onto the disc {z:z<1}\{z:|z|<1\}. [Hint: You may find it helpful to consider a mapping of the form (az+b)/(cz+d)(a z+b) /(c z+d), where ad bc0-b c \neq 0.]

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