ActualMidtermSol

# ActualMidtermSol - McGill University Department of...

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Unformatted text preview: McGill University Department of Mathematics and Statistics MATH381 Complex Variables and Transforms, Fall 2008 Midterm Exam Solutions 1. Let u ( x,y ) = e x ( x cos y- y sin y ). (a) Show that u is harmonic on R 2 . (b) Let f ( z ) be an analytic function whose real part is u . Compute f (- 1). Hint : It is possible to do this without computing the harmonic conjugate of u and without computing f . Solution: (a) u = e x ( x cos y- y sin y ) u x = e x ( x cos y- y sin y ) + e x cos y = e x ( x cos y- y sin y + cos y ) u xx = e x ( x cos y- y sin y + cos y ) + e x cos y = e x ( x cos y- y sin y + 2cos y ) u y = e x (- x sin y- sin y- y cos y ) u yy = e x (- x cos y- cos y- cos y + y sin y ) = e x (- x cos y + y sin y- 2cos y ) u xx + u yy ≡ Thus u is harmonic on R 2 . (b) Let f ( z ) = u ( x,y ) + iv ( x,y ). Then f z = u x + iv x and by the Cauchy-Riemann equations v x =- u y . Consequently, f z = u x- iu y and f (- 1) = e- 1 (- 1- 0+1)+ ie- 1 (--- 0) = 0. 2. Evaluate the following integrals. You have to name the theorems you’re using and you have to explain why they can be applied....
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ActualMidtermSol - McGill University Department of...

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