# hw10 - Math508, Fall 2010 Jerry L. Kazdan Problem Set 10 D...

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Math508, Fall 2010 Jerry L. Kazdan Problem Set 10 DUE: Tues. Nov. 30, 2010. Late papers will be accepted until 1:00 PM Wednesday . Note: We say a function is smooth if its derivatives of ball orders exist and are continuous. 1. Find an integer N so thst 1 + 1 2 + 1 3 + ··· + 1 N > 100. 2. Let c ( x ) be a given smooth function and u ( x ) ±≡ 0 satisfy the differential equation - u ²² + c ( x ) u = λ u on the bounded interval Ω = { a < x < b } with u = 0 on the boundary of Ω . Here λ is a constant. Show that λ = R Ω ( u ² 2 + cu 2 ) dx R Ω u 2 dx 3. The Gamma function is deﬁned by Γ ( x ) : = Z 0 e - t t x - 1 dt . a) For which real x does this improper integral converge? b) Show that Γ ( x + 1 ) = x Γ ( x ) and deduce that Γ ( n + 1 ) = n ! for any integer n 0. 4. Consider f ( x ) : = k = 1 sin kx 1 + k 4 . a) For which real x is f continuous? b) Is f differentiable? Why? 5. If the complex power series

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## This note was uploaded on 03/06/2012 for the course MATH 508 taught by Professor Staff during the Fall '10 term at UPenn.

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hw10 - Math508, Fall 2010 Jerry L. Kazdan Problem Set 10 D...

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