As part of a question to verify that , I have to use complex contour integration with the Bromwich contour to evaluate the inverse laplace transform of . We obviously know that the answer is but I'm not sure how to work it out. I don't know what contour to take...
I've tried taking the contour which goes like... and then considering the limit as but this just gives me 0 which is wrong. So I'm confused as to whether I've taken the wrong contour or have calculated this incorectly.
The hint says to take a contour similar to a keyhole...but alas I still cannot think of what it should be. Any help would be greatly appreciated - thanks in advance
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Last edited by lilman91; 14-04-2011 at 00:04.
- 14-04-2011 00:02
- 14-04-2011 09:06
Because sqrt(z) has a branch point, you need to put in a cut line so that sqrt(z) will be single valued on the region you're integrating.
As I understand it, (and as well as I can describe it on here), take a cutline along the -ve x (real) axis.
you then want a contour going from 1-iR to 1+iR, then a quarter semicircle radius R ending at (1-R, 0), then a straight line from (1-R) to (-eps, 0) (above the cutline), then a clockwise circle radius eps going to (-eps, 0) (below the cutline), then a straight line from (-eps, 0) to (1-R), and finally a quarter semicircle radius R going from (1-R, 0) to (1- iR) again to close the contour. Then let R go to infinity.
As I understand it, the two straight line integrals (from x=1-R to eps and back again) do NOT cancel out because sqrt(z) has a different value depending on whether you are above or below the cut.
Not sure how difficult this will be to actually do - it's been too long since I did Contour Integration in anger - but good luck!