Complex Analysis 22 | Goursat's Theorem

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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)

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

aleksandervadla

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

inthebackwiththerabbish

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

putin_navsegda

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.

guilhermefranco

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).

jagatiello

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…

SCPC

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

greggreen

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.

kamalsaleh

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