Complex Analysis 22 | Goursat's Theorem

Рет қаралды 13,183

Жыл бұрын

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This is my video series about Complex Analysis. I hope that it will help everyone who wants to learn about complex derivatives, curve integrals, and the residue theorem. Complex Analysis has a lof applications in other parts of mathematics and in physics.
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(This explanation fits to lectures for students in their first or second year of study: Mathematics, Mathematics for physicists, Mathematics for the natural science, Mathematics for engineers and so on)

Пікірлер: 22

@aleksandervadla9881 Жыл бұрын

One of my favourite theorems in complex analysis. Simpel, yet so elegant…

@inthebackwiththerabbish 5 ай бұрын

These videos are worth millions. Thank you so much for making them available to us for free

@brightsideofmaths 5 ай бұрын

Glad you like them! The videos are for free but it's worth to support me such that I can create more :)

@inthebackwiththerabbish 5 ай бұрын

@@brightsideofmaths I definitely will support you when I have the means to 👊

@guilhermefranco2949 2 ай бұрын

In Conway's Functions of One Complex Variable vol. 1, Chapter 4, Section 8 we see Goursat's theorem a little different. The proof in the video is the almost the same we see there, but we are doing it for another purpose: supposing f only differentiable (and not continuously differentiable) you want to show that the integral of f over any triangular path is zero to invoke Morera's theorem and conclude that f is holomorphic.

@meteor8076 Жыл бұрын

Very interesting, thanks !

@putin_navsegda6487 Жыл бұрын

Thank you for giving a proof without Green's Theorem!

@brightsideofmaths Жыл бұрын

Yes, I only wanted to use the stuff we already know. Otherwise, it just feels like magic.

@tonychio9207 18 күн бұрын

that so good

@jagatiello6900 Жыл бұрын

For those interested, there's a nice account on Contour Integration and Cauchy’s Theorem by Cosgrove (don't paste link bc YT erases it).

@brightsideofmaths Жыл бұрын

I don't know these videos but recommendations for other good resources to learn mathematics are always welcomed!

@jagatiello6900 Жыл бұрын

@@brightsideofmaths Most certainly...keep it up! Greetings from Rosario, Argentina.

@greggreen5510 5 ай бұрын

If we already know that the integral of a closed curve is zero, what does it matter that the image is a triangle?

@brightsideofmaths 5 ай бұрын

How do we know that? For example, the closed curved integral of 1/z along a circle around the origin is not zero.

@kamalsaleh6497 Жыл бұрын

7:58 But if it approaches a point, then aren't we supposed to finish the proof? Since I thought that integrating on a point gives zero and the only absolute value that is less than or equal to zero is also zero. Maybe my argument is not rigorous enough.

@brightsideofmaths Жыл бұрын

Note that we still integrate around the point.

@SC0PC Жыл бұрын

Why do we know in 9:26 that this part has an antiderivative? I understand this for f` but not for f(z_0). I thought we only know, that f has an Derivate…

@brightsideofmaths Жыл бұрын

Yeah, thanks for the question. It's a linear function. And we know that each polynomial has an antiderivative.

@SC0PC Жыл бұрын

@@brightsideofmaths Thank you very much. I get it now. :D BTW: This videos are very helpful particulary for my exams preparation. :) Thank you very much for your effort.

@brightsideofmaths Жыл бұрын

@@SC0PC I am very glad that my videos can help you :)