In this video, I begin my new series on Advanced Topics in Complex Variables with a lesson on deriving the Kramers-Kronig relations. The Kramers-Kronig relations allow the determination of the real part of a complex function from the imaginary part and vice-versa.
For this reason, they come up in many places in Physics that make use of complex variables. I begin this derivation by discussing the Cauchy Principal Value, setting up the assumptions, and performing an integration over a slightly modified semicircular contour. Then, I use the definition of the Principal Value, the polar representation of complex numbers, Jordan's Lemma, and Cauchy's Theorem to make some simplifications, after which I complete the derivation.
You'll notice that a lot of this proof takes elements from multiple videos in my Complex Variables playlist, so I highly recommend you watch those so that you can best engage with the content presented in this video!
Questions/Requests? Let me know in the comments!
Pre-reqs: My entire Complex Variables playlist: [ Ссылка ]
Improper Integral Video: [ Ссылка ]
Lecture Notes: [ Ссылка ]
Patreon: [ Ссылка ]
Twitter: [ Ссылка ]
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