Complex Analysis: Jordan's Lemma

Today, I present a proof for Jordan's lemma, a very useful result in complex analysis especially when calculating contour integrals. We use various estimations and inequalities to show that the value of the integral over a semicircular arc does indeed go to zero under certain conditions.

[prabalbaishya]

Nice! Also I think we could have used the fact that the exponential decay function converges extremely rapidly and when it is with some arbitrary function (which doesn't diverge very badly for example like e^z²) the limit as the independent variable approaches ∞ is 0. Here in our proof we have f(z) which we have restricted to as a rational function and I think there is a theorem which states that we can apply the limit, even before evaluating the integral. So we see that whatever divergence the linear term 'R' and |f(z)| produces is killed by the strong convergence of the e^-(aRsint) and as t lies between 0 and π so sint is always >0. So we can conclude that as R gets bigger and bigger the integral goes to 0. I might have done some mistakes while explaining as I have just started out complex analysis. Also your videos are just awesome for me!

[jackmoffatt]

Really good and clear explanation. Much appreciated :)

[wuyqrbt]

*Long time no see!!*
*I'm waiting for your next video*

[juniorcyans]

I wish you could have do some examples to show how to apply the theorem. That would be more helpful. Anyways, thank you very much!

[lucasfranco]

Could you move the limit as R goes to infinity inside of the integral instead of using an inequality?

[wuyqrbt]

Hello Dear *QN3* .
I Have a request, please speak about *Bessel functions* too.
Thank you

[abcdef]

at 18:35 what if a =0 then aR = undetermined, ....whenever you are integrating a big semi circle R....but most com integrals dont have exp(iaz)