MATH 227 Final, Spring 1997

Mathematics 227, Second Semester 1996-97 Final Examination, June 18, 1997, 8:00 - 10:30 Problem 1. 15 pts. Write everything you know about the exponential function f(z) = e*. Problem 2. 15 pts. Write what, you know about the classification of isolated singular points of an analytic function. Problem 3. 9 pts. Let z = «+ iy. Find the “if and only if” condition in terms of z and y for z? to be an imaginary number. Problem 4. 9 pts. Suppose that f(z) is differentiable at z). Write the proof that f satisfies the Cauchy- Riemann equations at. zo. Problem 5. 9 pts. Find the harmonic conjugate of f(x,y) = 2° — 3ry? — 3x. Problem 6. 9 pts. Evaluate the following integral: no; a) fz 2 coszdz. JO Problem 7. 9 pts. Prove that the series S> as) z” n=l fz converges absolutely for each z such that |z| < 1, and diverges when |z| > 1. Problem 8. 9 pts. Write the Maclaurin series of f(z) = What is its radius of convergence? Prohlem QQ pts. Suppose that f(z) is analytic in a domain D such that 0 € D. Let S3°5 enz” be the Maclaurin series of f. Prove that if f(z) = —f(—z) for each z € D, then co, = 0 for each k=0,1,2,... Problem 10. 9 pts. Suppose that a,b,c€ R and ac—b? > 0. Use residues to find the value of the following . ale sO dr integral: Joe az2+Jbe+e * anne TY | \ Good Luck!