Partial Differential Equations and Applied Mathematics
Partial Differential Equations and Applied Mathematics MATH 5220
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This 1 page Class Notes was uploaded by Ila Haag on Monday September 21, 2015. The Class Notes belongs to MATH 5220 at University of Virginia taught by Staff in Fall. Since its upload, it has received 16 views. For similar materials see /class/209583/math-5220-university-of-virginia in Mathematics (M) at University of Virginia.
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Date Created: 09/21/15
Math 522 Take home exam You may use your books and notes but no discussing among each other Please sign honor pledge Explain your work The exam is due in class Feb 26th 1 15 points Suppose that is a function analytic in the entire complex plane C and suppose moreover that f satis es a bound fz S 01 lzl 4 for some constant c and for all z E C The problem is to show that in fact f is constant a Differentiate the Cauchy integral formula 1 f 5 d5 3 i Z for 2 inside the region bounded by T to obtain an integral formula for f Do this repeatedly to obtain an integral expression for f z the nth derivative of f b Use your formula from a the given bound on f above and T a circle of radius R with R gt 00 to show that fO 0 Explain estimates on your contour integral c Show that in fact f 0 0 for all n 2 1 Since f has an entire power series this shows that f is constant representing f by its Taylor series 2 10 points Determine the sum in closed form 1 n20 a2 7127