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gangsta316
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http://www2.imperial.ac.uk/~bin06/M2...nation2008.pdf
Solutions are here.
http://www2.imperial.ac.uk/~bin06/M2...insoln2008.pdf
My first question is about 3(ii), the proof of Cauchy's integral formula for the first derivative.
The proof here uses the deformation lemma
(from second page here:
http://www2.imperial.ac.uk/~bin06/M2...pm3l18(11).pdf )
and proves the theorem for an approximating contour. I made up the proof myself using the ideas from what we were taught (so I remembered the gist of the proof, not all of it) and I think that I made one without the use of this lemma. Why is it needed? Can we not just say that, since the interior of g (g for gamma) is open, a+h is inside g for |h| small enough. Then we can write f(a+h) as a contour integral around g and then take the difference quotient and let h->0. Is this ok or do we need to be working in a circular contour for the proof to work? I always thought something was not quite right about how he just let's h->0 inside the integrand like that (i.e. without evaluating the integral) -- is that what requires the contour to be a circle?
4(ii).
Would it be correct to substitute x = tany and change the limits to -pi/2, +pi/2? In general how do we know that these will be the limits for this integral? tan also blows up at (pi/2 + 2*pi) so why did we choose pi/2 for the upper limit?
Thanks for any help.
Solutions are here.
http://www2.imperial.ac.uk/~bin06/M2...insoln2008.pdf
My first question is about 3(ii), the proof of Cauchy's integral formula for the first derivative.
The proof here uses the deformation lemma
(from second page here:
http://www2.imperial.ac.uk/~bin06/M2...pm3l18(11).pdf )
and proves the theorem for an approximating contour. I made up the proof myself using the ideas from what we were taught (so I remembered the gist of the proof, not all of it) and I think that I made one without the use of this lemma. Why is it needed? Can we not just say that, since the interior of g (g for gamma) is open, a+h is inside g for |h| small enough. Then we can write f(a+h) as a contour integral around g and then take the difference quotient and let h->0. Is this ok or do we need to be working in a circular contour for the proof to work? I always thought something was not quite right about how he just let's h->0 inside the integrand like that (i.e. without evaluating the integral) -- is that what requires the contour to be a circle?
4(ii).
Would it be correct to substitute x = tany and change the limits to -pi/2, +pi/2? In general how do we know that these will be the limits for this integral? tan also blows up at (pi/2 + 2*pi) so why did we choose pi/2 for the upper limit?
Thanks for any help.
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