{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

MA530_JAN08

# MA530_JAN08 - 4 ²ind a fractional linear transformation...

This preview shows pages 1–2. Sign up to view the full content.

Math 530 Qualifying Exam, January 2008 Professor: A. Eremenko All problems have equal weight, and it is split equally between a) and b) if there are a) and b). In your proofs, you can use any theorem stated in class provided that you state it completely and correctly. Name: 1. Suppose that a function u , harmonic in a neighborhood of the origin, equals to zero on the real and imaginary axes. Prove that u is even. 2. a) Prove that the series f ( z ) = summationdisplay n =0 2 n 2 z 2 n converges uniformly in the closed unit disc, and that the limit function is infinitely differentiable in the closed unit disc and analytic in the open unit disc. b) Prove that this function f is not analytic at any point of the unit circle, that is the radius of convergence of the series summationdisplay n =0 f ( n ) ( z 0 ) n ! ( z - z 0 ) n is zero for every z 0 on the unit circle. 3. Evaluate the integral integraldisplay 0 sin(Log x ) x 2 + 4 dx. Here Log is the principal value of the logarithm (real on the positive ray).

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 4. ²ind a fractional linear transformation that maps the two circles { z : | z | = 1 } and { z : | z-1 | = 3 } onto some concentric circles. 1 5. a) Prove that every function of the form az + b-m s n =1 c k z-t k , (1) where b and t k are real, a ≥ 0 and c k > 0, maps the upper half-plane into itself and the lower half-plane into itself. b) Prove that every rational function that maps the upper half-plane into itself and the lower half-plane into itself has the form (1). 6. Let f be an analytic function in a neighborhood of 0, that satisFes the functional equation f ′ ( z ) = qf ( q 2 z ) , f (0) = 1 , with some q, | q | ≤ 1. ±ind explicitly the Taylor coe²cients of f at 0 and then determine the radius of convergence of its Taylor series. 2...
View Full Document

{[ snackBarMessage ]}

### What students are saying

• 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.

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

• 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.

Dana University of Pennsylvania ‘17, Course Hero Intern

• 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.

Jill Tulane University ‘16, Course Hero Intern