Complex Analysis 31 | Application of the Identity 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.

This is the last video in the Complex Analysis Series.

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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)

[synaestheziac]

So, is the result in this video the same thing as analytic continuation, or am I mixing things up?

[RSLT]

Great video. Thank you very much. Any video regarding the uniqueness of analytic continuation?

[StratosFair]

I'm loving this series so far, thank you so much for your work !!

[narfwhals]

Knowing the identity theorem Analytic Continuation seems basically trivial.
All other sources I have found make it seem much more complicated than "find the powerseries. You're done".
Why do we have the extra steps of going through many regions of convergence? Is this to find a path connected domain?

[ferry]

can you explain a bit more why if D is open set, then accumulation point will be in D?

[whatitmeans]

I have a question about the example: It is a requirement to f(x) to be a smooth function?
As example, think in this function
f(x)=1/4*(1-x/2+|1-x/2|)^2
Does the theorem still holds?

[RSLT]

Great Video. Following my comments on your last video. So I like to repeat my question. How can I say this in exact math terms: Assume zeta (s)=zeta(1-s). Therefore because zeta (s)-zeta(1-s) =0 the Sum 1/n^s - Sum 1/n^(1-s) or sum(( 1/n^s) -1/n^(1-s)) converge to zero in the critical strip. I want to use the identity theorem. The challenge is power series 1/n^s is a divergence in the critical strip. Please advise