609S04ex1short - Math 609 Exam 1 Jerry L Kazdan 1:30 2:50...

This preview shows page 1 out of 1 page.

Math 609 Exam 1 Jerry L. Kazdan March 18, 2004 1:30 — 2:50 Directions This exam has two parts, the first is short answer ( 6 points each ) while the second has traditional problems ( 12 points each ). Closed book, no calculators – but you may use one 3” × 5” card with notes. Part A: Short Answer Problems (5 problems, 6 points each) A–1. Find all complex values of 1 i in the form a + ib . A–2. a) For which values of the constant c is u ( x, y ) := 2 y + e 3 y sin cx the real part of an analytic function f = u + iv ? b) For these values of c , find the corresponding function f ( z ). A–3. If a n z n is the power series expansion of 1 cos( z + 1) about z = 0, what is its radius of convergence? A–4. Compute C e z z 2 - 2 z dz , where C is the ellipse x 2 / 25 + y 2 / 9 = 1 (counterclockwise). A–5. Let f ( z ) be holomorphic for 0 < | z | < . If f has no zeroes and | f ( z ) | ≥ | f (2) | in the disk | z - 2 | < 1, what can you conclude about f ( z )? Justify your assertions. Part B: Traditional Problems (6 problems, 12 points each) B–1. Let f ( z ) be holomorphic in {| z
Image of page 1

Unformatted text preview: | ≤ 1 } except for a simple pole at z = i/ 2. If f also satisfies f ( 1 2 ) = 0 as well as | f ( z ) | ≤ 1 on | z | = 1, show that | f (0) | ≤ 1. B–2. Find a conformal map f ( z ) = u + iv from the unit disk {| z | < 1 } to the first quadrant, { u > , v > } . B–3. Give a complete clear proof of the fundamental theorem of algebra: Every nontrivial complex polynomial has at least one root. B–4. Evaluate Z ∞ cos x 1 + x 2 dx . B–5. Let g ( z ) be holomorphic in the closed unit disk D = {| z | ≤ 1 } and assume that | g ( z ) | ≤ 2 for | z | = 1. How many roots does h ( z ) := g ( z ) + 5 z 3-2 have in D ? As usual, justify your assertions. B–6. Let h ( z ) = ∞ X n =1 a n n z , where z = x + iy , and assume the sequence a n is bounded, say | a n | ≤ M . Show that h ( z ) is holomorphic in the half-plane { x > 1 } ....
View Full Document

{[ snackBarMessage ]}

Get FREE access by uploading your study materials

Upload your study materials now and get free access to over 25 million documents.

Upload now for FREE access Or pay now for instant access
Christopher Reinemann
"Before using Course Hero my grade was at 78%. By the end of the semester my grade was at 90%. I could not have done it without all the class material I found."
— Christopher R., University of Rhode Island '15, Course Hero Intern

Ask a question for free

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern