Uploaded April 2019 | Updated September 2026, 2 weeks ago
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: youtube.com/playlist?list=PLdgVBOaXkb9CNMqbsL9GTWwU542DiRrPB
Improper Integral Video: youtube.com/watch?v=a2ocry6LBe0
Lecture Notes: drive.google.com/open?id=1dxk08qUIfItegpJwG2FXww8fF0Ozj7yt
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan
Special thanks to my Patrons for supporting me at the $5 level or higher:
- Anonymous
- Cesar Garza
- Odissei
- Alvin Barnabas
- Jacob Soares
- Yenyo Pal
- Lisa Bouchard
- Bernardo Marques
- Connor Mooneyhan
- Richard McNair
- Guillaume Chereau
- Patapom
- Vitor Ciaramella
- McKay Oyler
- Dieter Walter Reule
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: youtube.com/playlist?list=PLdgVBOaXkb9CNMqbsL9GTWwU542DiRrPB
Improper Integral Video: youtube.com/watch?v=a2ocry6LBe0
Lecture Notes: drive.google.com/open?id=1dxk08qUIfItegpJwG2FXww8fF0Ozj7yt
Patreon: patreon.com/user?u=4354534
Twitter: twitter.com/FacultyOfKhan
Special thanks to my Patrons for supporting me at the $5 level or higher:
- Anonymous
- Cesar Garza
- Odissei
- Alvin Barnabas
- Jacob Soares
- Yenyo Pal
- Lisa Bouchard
- Bernardo Marques
- Connor Mooneyhan
- Richard McNair
- Guillaume Chereau
- Patapom
- Vitor Ciaramella
- McKay Oyler
- Dieter Walter Reule
