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

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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 Cauchy–Riemann equations to find their imaginary parts and hence deduce the forms of the complex functions.hence deduce the forms of the complex functions....
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exsheet4 - Mathematical Methods I Natural Sciences Tripos...

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