Complex analysis to solve integrals.

OK I have experience with complex analysis and have understood it well at some point (it goes easily) , but at this point I just want some basic cookbook rules at how this stuff works. Specifically, when solving an integral and using a contour in the complex plane, is there a quick and easy way to tell if it dissapears? I know it has to do with Jordan's lemma, but is there a way you can pretty much instantly tell? My aim is to just be able to solve complex analysis problems quickly, because for me its a tool and I am not interested in learning the maths in detail. Also I take it in general you use some kind of semi circular contour? Perhaps one that avoids poles on the real axis.

Any guidance would be much appreciated.

Reply 1

TeeEm

19

Original post by QuantumOverlord

OK I have experience with complex analysis and have understood it well at some point (it goes easily) , but at this point I just want some basic cookbook rules at how this stuff works. Specifically, when solving an integral and using a contour in the complex plane, is there a quick and easy way to tell if it dissapears? I know it has to do with Jordan's lemma, but is there a way you can pretty much instantly tell? My aim is to just be able to solve complex analysis problems quickly, because for me its a tool and I am not interested in learning the maths in detail. Also I take it in general you use some kind of semi circular contour? Perhaps one that avoids poles on the real axis.

Any guidance would be much appreciated.

I wrote those 2 booklets last summer for use by engineers and physicists.

http://madasmaths.com/archive/maths_booklets/advanced_topics/residues_and_applications_in_integration.pdf

and for summing series

http://madasmaths.com/archive/maths_booklets/advanced_topics/residues_and_applications_in_series_summation.pdf

maybe they are useful (although I may be going into too much analytical detail)

Reply 2

davros

16

Original post by QuantumOverlord

OK I have experience with complex analysis and have understood it well at some point (it goes easily) , but at this point I just want some basic cookbook rules at how this stuff works. Specifically, when solving an integral and using a contour in the complex plane, is there a quick and easy way to tell if it dissapears? I know it has to do with Jordan's lemma, but is there a way you can pretty much instantly tell? My aim is to just be able to solve complex analysis problems quickly, because for me its a tool and I am not interested in learning the maths in detail. Also I take it in general you use some kind of semi circular contour? Perhaps one that avoids poles on the real axis.

Any guidance would be much appreciated.

I think the point is that you can't "instantly tell". Jordan's Lemma will help with certain classes of integral where you're using a semicircular contour (or possibly a wedge shape), but there are some functions where you're going to need to be more inventive - e.g. using a rectangular contour or one with indentations.

You can't "avoid" poles on the real axis if the function you're ultimately trying to integrate actually has poles on the real axis! So the answer to your question is really "it depends".

If you want a book that focuses on the calculations rather than a lot of theory then there is an excellent book in the Schaum's Outline series. Also things like Engineering Mathematics by K A Stroud (possibly the Advanced version) cover stuff like this from a practical point of view.