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*URGENT* (Exam tomorrow morning!) Calculating a real integral using residue theorem
See attached picture
http://imageshack.us/photo/my-images/827/unledozs.jpg/
Parts a) and b) are straightforward.
For b) I end up with (using the residue theorem) 2.pi.i (\frac{1}{2.2^{1/2}}(1+i) + \frac{1}{2.2^{1/2}}(1-i))
= 2.i.pi.2^{1/2)}
I have used I to represent the real integral the question is asking us to evaluate.
for c), I can show that the contributions from the two circles on the contour are both 0, but the contributions from horizontal line just above the real axis is I and from the line just below the real axis is -I*exp(i.pi)=I, so we end up with I = i.pi/root(2). This is, of course, wrong: it is a real integral, it makes no sense for the answer we get to be imaginary. I'm certain my answer for b) is correct, so please could someone talk me through the procedure for relating the contour integral to I please?
Homework Statement
See attached picture
http://imageshack.us/photo/my-images/827/unledozs.jpg/
Homework Equations
The Attempt at a Solution
Parts a) and b) are straightforward.
For b) I end up with (using the residue theorem) 2.pi.i (\frac{1}{2.2^{1/2}}(1+i) + \frac{1}{2.2^{1/2}}(1-i))
= 2.i.pi.2^{1/2)}
I have used I to represent the real integral the question is asking us to evaluate.
for c), I can show that the contributions from the two circles on the contour are both 0, but the contributions from horizontal line just above the real axis is I and from the line just below the real axis is -I*exp(i.pi)=I, so we end up with I = i.pi/root(2). This is, of course, wrong: it is a real integral, it makes no sense for the answer we get to be imaginary. I'm certain my answer for b) is correct, so please could someone talk me through the procedure for relating the contour integral to I please?
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