Math 520a - Final take home exam - solutions 1. Let f (z) be entire. Prove that f has finite order if and only if f has finite order and that when they have finite order their orders are the same. Solution: Suppose that f satisfies |f (z)| A exp(B|z| ) By
Math 520a - Midterm take home exam Do 5 out of the 6 problems. Do not turn in more than 5. The exam is due Monday, Nov 2 at the start of class. Late papers will only be accepted in the case of illness. You may consult the textbook, your class notes and ot
Math 520a - Homework 6 - selected solutions 1. Let be an open subset of the complex plane that is symmetric about the real axis and intersects the real axis. Let + = cfw_z : Im(z) > 0. Let I = cfw_z : Im(z) = 0, the intersection of with the real axis. Let
Math 520a - Homework 5 - Selected solutions 1. Is it possible to define a branch of the logarithm f (z) such that for all positive integers n, f (n) = log(n) + 2in ? You should justify your answer, i.e, show it cannot be done or show how to do it. Solutio
Math 520a - Homework 4 - Selected solutions 1. Problem 5 on page 103 in the book. Solution: I'll just make a comment on this one. You close the contour with a semicircle in either the upper or lower half plane depending on the sign of . Most people worked
Math 520a - Homework 3 1. For each of the following four functions find all the singularities and for each singularity identify its nature (removable, pole, essential). For poles find the order and principal part. Solution: z cos(z -1 ) : The only singula
Math 520a - Homework 2 1. Use Cauchy's integral formula (for an analytic function or its derivatives) to evaluate (a) For the contour (t) = eit , 0 t 2, the integral eiz dz z2
(b) For the contour (t) = 1 + 1 eit , 0 t 2, the integral 2 ln(z) dz (z - 1)n
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Math 520a - Homework 1 1. Let f be analytic on a region (a connected open set). Prove that in each of the following cases f is a constant. (a) f () is a subset of the real line. (b) f () is a subset of some line. (c) f () is a subset of some circle. Solut
Math 520a - Final take home exam - version 1 Do 5 out of the 6 problems. Do not turn in more than 5. You may work on the exam for one continuous week during the period Mon, Dec 7 to Fri, Dec 18. The exam is due at 3 pm on Fri, Dec 18 at the latest. Late p
Math 520a - Midterm take home exam Do 5 out of the 6 problems. Do not turn in more than 5. The exam is due Monday, Nov 2 at the start of class. Late papers will only be accepted in the case of illness. You may consult the textbook, your class notes and ot