Morera's Theorem in Practice - Laplace Transform

Here we examine an application of Morera's Theorem in complex analysis to show the analyticity of the Laplace transform.

What sort of function is the transform of a signal in the frequency domain? To be perfectly honest, when I first learned it, I didn't think that Morera's theorem would have been very useful. It says that if you have a function that is continuous and it's integrals along each triangle in the domain of the function is zero, then the function is analytic. I thought how in the world would you know those two conditions without already knowing that a function was analytic.

Well, that's egg on my face, because, Morera's theorem helps a lot. In fact, whenever you have a transform like the Laplace tranform, it really comes in to save the day, and this all comes down to the "basis" of the Laplace transform being e^{-st}, or entire.

The smoothness and analyticity of the Laplace transform is leveraged throughout systems theory and control. In your ordinary differential equations class, every example that is presented is not only analytic (or meromorphic) but is also rational. This is really handy when you have to invert a Laplace transform using tables, where a partial fraction decomposition breaks down a Laplace transform of a solution to a differential equation into a collection of pieces that you can match up easily.

However, analytic functions are much more than just rational functions, and so the question arises as to what functions (or signals) give rise to an analytic Laplace transform. The resolution to this question is the content of the Pailey Wiener theorem (or well, one of them), where an L2 signal may be extracted as the inverse of a function that is analytic in the right half of the complex plane.

What about the other way around? If we have an L2 signal, can we show that its Laplace transform is analytic in the right half plane? The answer, perhaps unsurprisingly, is yes. And the proof of this comes down to a touch of the Dominated Convergence Theorem and Morera's theorem.

If you are curious to see Morera's Theorem in action, then check out this video, which is Part 1 of two laying out the Pailey Wiener Theorem.

//Additional Reading
Zemanian - An Introduction to Generalized Functions and the Generalized Laplace and Legendre Transformations

//Exercises

//Watch Next

//Music Provided by Epidemic Sound
Mt. Fuji - Ooyy
Vegas - Onda Norte
bam. bam. - Yomoti
Tiny Things - Giants' Nest
Late Nights - Daxten

Use this referral link to get a 30 day free trial with Epidemic Sound for your YouTube channel:

//Books

//Recording Equipment

DISCLAIMER: The links above in this description may be affiliate links. If you make a purchase with the links provided I may receive a small commission, but with no additional charge to you :) Thank you for supporting my channel so that I can continue to produce mathematics content for you!

[JoelRosenfeld]

Have you ever learned a theorem, thought it would be useless, then ended up using it all the time?

[michaelwirtzfeld]

I liked the original production of your videos. No background music and a set camera position. The new format, for me, gets in the way of your content.

[TriThom]

Can you more rigorously define what you mean for the laplace transform to be analytic in the RHP? Like how do we know the fact that it’s continuous and the fact it’s integral is 0 along arbitrary triangles makes it analytic and why do we care?

[fanalysis]

So moreras theorem had a use after all