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May 7th, 2012, 01:45 AM  #1 
Newbie Joined: May 2012 Posts: 2 Thanks: 0  Conditions for convergence of real integral
Hi Everybody, I've been pondering this question for a while now, and have realised (after far too long...) that I really need some help. Here's the question: Consider a real function f(x). How do the conditions for the convergence of the real integral of f(x) relate to the conditions under which the integral of f(z) goes to zero on C_R? (Here C_R is the simple closed contour described counterclockwise consisting of: the segment of the real axis from z=R to z=R, and the top half of the circle abs(z)=R.) I know the question is somewhat ambiguous (it makes no suggestion as to the type of real integral of f(x): improper? definite? indefinite?), so I have only been considering the case of improper real integrals of f(x) so far. For this particular case, I have considered: 1. the cauchy principle value of an improper real integral of f(x), 2. using residue theory to rewrite a real integral of f(x) as int[f(x), R, R]+int[f(z), C_R] = i2pi*Res(f(z), z=z_k), 3. showing that if R is made large enough, the contour mentioned above will contain the singularities of f(z) that lie in the upper half of the complex plane (y>=0), and 4. the value of int[f(z), C_r] will go to zero as R goes to infinity, and so int[f(x), R, R] = i2pi*Res(f(z), z=z_k). From this, I have so far concluded that if the improper real integral of f(x) converges, then: 1. f(z) has no singularities that lie on the real axis, and 2. the value of the integral int[f(z), C_R] must go to zero as R goes to infinity. However, I do not know if this approach is even answering the question (in the case of improper real integrals of f(x)). Am I on the right track?? Once again, so far I have only considered improper real integrals of f(x). Any help would be very, very much appreciated, thanks everybody!! 
May 7th, 2012, 03:20 AM  #2 
Newbie Joined: May 2012 Posts: 2 Thanks: 0  Re: Conditions for convergence of real integral
Edit to previous post: C_R is the simple closed contour described counterclockwise consisting of the top half of the circle abs(z)=R. It does not include the segment of the real axis from z=R to z=R. My apologies, 

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