homework20111005

homework20111005 - Math 417, Fall 2011 Exercise Set #5 = i...

Info iconThis preview shows page 1. Sign up to view the full content.

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

Unformatted text preview: Math 417, Fall 2011 Exercise Set #5 = i to z = 1. Show that 4 2. Turn in: October 5 1. Let C denote the line segment from z dz 4 C z 2. Let CR denote the upper half circle for |z| = R, for R > 1, parametrized in the counter-clockwise direction. Show that Log(z) + ln R dz 2 . 2 z R CR Conclude that the integral tends to zero as R tends to infinity. 3. Suppose that f (z) and g(z) are analytic functions defined in a domain , and that C is a contour in starting at z1 and ending at z2 . Show that: f (z) g (z) dz = f (z2 )g(z2 ) - f (z1 )g(z1 ) - g(z) f (z) dz C C For the next three problems, all integrals can be done using results from class or the book, and without parametrizing the contour C. 4. Calculate the following integrals, where C is the positively oriented boundary of the square with vertices at 2 - 2i, 2 + 2i, -2 + 2i, -2 - 2i. e-z a) dz C z -i z2 + 8 b) dz C 2z - 1 5. Calculate the following integrals, where C is the positively oriented circle |z-i| = 2. dz a) dz 2+4 C z dz b) dz 2 2 C (z + 4) 6. Calculate the integral 0 x4 dx . x8 + 1 1 ...
View Full Document

This note was uploaded on 02/27/2012 for the course MATH 417 taught by Professor Staff during the Fall '11 term at SUNY Albany.

Ask a homework question - tutors are online