Complex Analysis L13: Bromwich Integrals and the Inverse Laplace Transform

This video is a culmination of this series on complex analysis, where we show how to compute the Bromwich integral used in the inverse Laplace transform.

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This video was produced at the University of Washington

[gean]

It's unbelievable an amazing course like this available completely free on YouTube. The guy is really good!

[tolkienfan]

I really appreciate you making these lectures public. It's dense material, but that's exactly what I'm looking for. Thank you

[guiliangzheng]

What a ride in the complex analysis world! Thank you so much for putting it together! What a ride it is

[ElMalikHydaspes]

Bravo! What an excellent set of lectures on complex numbers! Really well taught by Dr Brunton.

[timepass]

thanks a lot, completed the whole complex analysis 10 hours before my finals, You're a brilliant teacher!!!

[andresfeliperamirezgaviria]

Excelent course, greetings and congratulations.

[basics]

How many hours do you have in one day? 70? More than 70?

I just scrolled xN speed (with N huge) this series about complex analysis. Very well done.

Lots of students in Engineering dealing with dynamical systems and control (so, almost every student in Engineering) curious about some detail about the math behind them and coming across these lectures should be so thankful to you. Obviously they're not enough without personal effort and study, but they're a good point to start for sure.

Anyone who wants a concise and quite precise introduction to complex analysis and many other mathematical topics useful in engineering, could have on Schaum's Outlines, Advanced Mathematics for Engineers and Scientists: 10-15 pages of theory for every topic, and proofs left as an exercise to the reader.

[danielhoven]

Thank You! I just finished a dynamics homework with no reference to a Laplace transform table. Un-necessary for sure, but I feel like a boss lol. Anyway the only bit I had to dig for myself was finding residues of higher order poles, but without your introduction I'd have struggled to make sense of the literature.

[andrej]

I believe those "tricks" for showing that parts of integral in complex plane are 0 etc...are called Jordans lemmas (theorems)....unfortunately I do not have my textbook with me and it has been over 20 years...

[eig_himanshu]

superb you way of explanation is fantabulous

[papawhiskeybravo]

Thanks for a great series. I was very well taught.

[tariqandrea]

This isn't mere mathematics. It is a work of community service, a work of kindness, and a work of charity.

[hoseinzahedifar]

Great great great lecture...Thank you so much.

[leonardoalcayaga]

Your lectures are really great, thanks a lot! Is it possible to follow a similar apporach to obtain Fourier transforms?

[byronwatkins]

At 34:50, the way theta and the contour are defined requires integration from pi/2 to -pi/2; but the limit is still zero. Jordan's lemma proves this in general...

[pkiwan]

Good lecture. Thank you for your lecture.
However, I still have a question that makes me unhappy. Originally, f(s) is defined in s > gamma (Laplace transform). In the region of s < gamma, f(s) diverges so that f(s) itself is not defined there. However, this contour uses this diverging area. How can this be solved? Some textbooks explain that analytic continuation justifies it. However, this does not seem to solve the paradox. Is it possible because exp(st) converges to zero faster than f(s) diverges?

[ralvarezb]

minute 41:00 I think the trick is |R-a| = sqrt((R-a)^2) wich leads to sqrt(R^2 - 2aR + a^2) If theta = pi, the inequality becomes equal, so true, but if theta is different from pi, the term 2aR cos(theta) < 2aR which is also true. Then you're right

[kovub]

Hey Steve, I really like your videos, and I'm curious – are you writing in reverse or did you flip the image? Either way, it's a cool effect!🤔

[xenofurmi]

ATHF reference in last video of a Complex Analysis lecture... the future is now!

[marekw]

Thank you for the course, I appreciate the conceive form.

You've been mentioning, that in good ol' days there would have been a whole semester course on complex analysis. Could you maybe recommend any sources to dive deeper into the topic?