.. 1990 Mar 23 .. =========== Examination 2 ============= 1990 Mar -------- **Instructions.** There are 5 questions: Each question will be graded on a scale of 0-10. A passing score at the Ph.D. level is 30. A passing score at the M.A. level is 20. **Notation.** The symbol :math:D stands for the unit disc :math:\{z \in \mathbb C \mid | z | < 1\}. .. ---------------- .. index:: Riemann mapping theorem .. _1990May_1: .. proof:prob:: (a) State the Riemann mapping theorem (including the relevant topological definitions). (b) Present a conformal mapping :math:f of :math:D onto :math:\Omega = \{z= x + i y \mid x > 0, y > 0\} such that :math:f(0)= 1 + i. .. ---------------- .. index:: residue theorem .. _1990May_2: .. proof:prob:: Evaluate :math:\int_{-\infty}^\infty \frac{dx}{1+x^2} by use of the residue theorem. .. ---------------- .. _1990May_3: .. proof:prob:: Suppose that :math:f is holomorphic on :math:D, :math:|f(z)| < 1 for all :math:z\in D, and :math:f(1/2) = 1/2. (a) Prove that :math:|f'(1/2)|\leq 1. (b) Determine :math:f(z) if :math:f'(1/2) = 1. .. ---------------- .. _1990May_4: .. proof:prob:: Let :math:\{f_n(z)\} be a sequence of nonvanishing analytic functions on a region :math:\Omega. Suppose that :math:\{f_n(z)\} converges uniformly on every compact subset of :math:\Omega to a function :math:f(z). If :math:f(z_0) \neq 0 for some :math:z_0 \in \Omega, then prove that :math:f(z)\neq 0 for all :math:z\in \Omega. .. ---------------- .. _1990May_5: .. proof:prob:: Suppose that :math:U(z) is a real valued harmonic function defined for all :math:z \in \mathbb C. If .. math:: \lim_{y\to 0+}\frac{u(x+iy)}{y} = O for all :math:x \in \mathbb R, then prove that :math:u(z) = 0 for all :math:z \in \mathbb C. ---------------------------- Solutions --------- .. .. container:: toggle .. .. container:: header Solution to :numref:Problem {number} <1990May_1> (coming soon) .. .. container:: toggle .. .. container:: header Solution to :numref:Problem {number} <1990May_2> (coming soon) .. .. container:: toggle .. .. container:: header Solution to :numref:Problem {number} <1990May_3> (coming soon) .. .. container:: toggle .. .. container:: header Solution to :numref:Problem {number} <1990May_4> (coming soon) .. .. container:: toggle .. .. container:: header Solution to :numref:Problem {number} <1990May_5> (coming soon) --------------------------------- .. insert space here