Complex Analysis 23 | Cauchy'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)

[hyperduality]

Integration is dual to differentiation.
Anti-derivatives are dual to derivatives -- holomorphism.
Convergence (syntropy) is dual to divergence (entropy).
"Always two there are" -- Yoda.

[extraflash]

When I took complex analysis I remember we also proved this theorem for when the function is holomorphic in D\{z0} and continuous in D, we did this with 2 sub proofs: first proved that if f is holomorphic in U\{z0} where U is an open set, and continuous in U, then for every rectangle in U the line integral is zero. And if a function is continuous in a disk D and for every rectangle (which is parallel to the axis) the line integral is zero, than f is a derivative of another function, and than the line integral of every closed curve must be zero.

[NoanMoysu]

Thank you for your videos, they are amazing! I have one question, at 5:46 was it also possible to pick the straight line from z_0 to z as the curve gamma_z and the straight line from z_0 to z_tilde as the curve gamma_z_tilde? I'm asking because I don't see a reason why it wouldn't work and it seems a bit simpler than the choice made in the video.

[GeoffryGifari]

in doing a contour integral, even when the closed contour crosses itself (Forming a figure 8, for example) it doesn't matter?

[lucasciacovelli]

Theres one point that i dont understand yet and thats at 8:04 how do you write the function as a contour integral? I cant prove to myself why that is True.

[malawigw]

This is my fav formula in all of mathematics!

[chanwoochun]

Thank you so much for all the videos. Question: Which step of the proof requires that D should be an open disc? The proof doesn't seem to require anything about the shape of D.

[guohaoyang]

“If the curve is a circle, we cannot conclude that the curve integral is 0 yet” Why we cannot say the curve integral is 0? And why we still need to prove Goursat’s and Cauchy’s theorem since we already have the theorem to ensure the integral on a closed curve is 0.
For the first question, does that sentence mean we cannot conclude that the curve integral over a circle is 0 from curve integral over a triangle?