exsheet4 - Mathematical Methods I Natural Sciences Tripos,...

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

View Full Document Right Arrow Icon

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

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

Unformatted text preview: Mathematical Methods I Natural Sciences Tripos, Part IB Dr Gordon Ogilvie Michaelmas Term 2008 Example Sheet 4 1. Define the circle of convergence of a power series of a complex variable z . Find the radii of convergence of the three series summationdisplay n =0 nz n , summationdisplay n =0 (cosh n ) z n and summationdisplay n =0 [2 n + ( 1) n ] z n . Show by example that a power series may or may not converge on its circle of convergence. Hence give an example of a series that is convergent but not absolutely convergent. 2. Calculate the Taylor series of the function f ( z ) = ln(1 z ) about z = 0 and determine its radius of convergence. Now calculate the Taylor series for f ( z ) about z = i and determine the new radius of convergence. Comment. 3. The real parts of three analytic functions of z = x + i y are sin x cosh y , e y 2 x 2 cos 2 xy , x x 2 + y 2 , respectively. Use the CauchyRiemann equations to find their imaginary parts and hence deduce the forms of the complex functions.hence deduce the forms of the complex functions....
View Full Document

Page1 / 2

exsheet4 - Mathematical Methods I Natural Sciences Tripos,...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online