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Mathematics 185 – Intro to Complex Analysis Fall 2005 – M. Christ Problem Set 11 Please ﬁnish reading Chapter 12. Our emphasis in this chapter is on applying the theory we’ve developed to concrete calculations and problems. We’ll do still more problems from this chapter in a future problem set. Due Tuesday November 22: Solve these problems from Chapter 12: 5(iv,v), 11,12,13, 16(iii), 17(i), 19. Hints : 5(v): You may use without proof the related fact that R ∞ 0 log( x ) 1+ x 2 dx = 0 (the substitution x = 1 /y expresses this integral as minus one times itself!). Method (a): Substitute x = e t to get an integral over the whole real line. Then apply the method we used in class to compute R ∞ -∞ e ax 1+ e 2 x dx . Note that the “related fact” mentioned above translates to R ∞ -∞ te t 1+ e 2 t dt = 0. Method (b): Let Log denote the branch of the complex log function deﬁned by Log( re iθ ) = ln( r ) + iθ for - π 2 < θ < 3 π 2 . Express R 0 -∞ Log( x ) 2 1+ x 2 dx in terms of the
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This note was uploaded on 07/31/2009 for the course MATH 185 taught by Professor Lim during the Spring '07 term at University of California, Berkeley.
- Spring '07