Cauchy’s integral formula is a classic and important result from complex analysis. Cayley-Hamilton is a classic and important result from linear algebra!
Would you believe me if I said that the first implies the second? That Cauchy implies Cayley-Hamilton is an extremely non-obvious fact, considering that the two are generally viewed as completely distinct subject matters.
I’m talking about high level videos of tactical insights and tips for good players to become even better. I’m not talking about channels that give obvious tips like ‘check your shoulder’ or ones that offer pointless skill moves.
I have an upcoming test in Complex Analysis for which I have been studying heavily. I have begun by memorizing the necessary definitions and theorems. I plan on following this by proving these major theorems and then proving as many practice problems as I can.
I was wondering if you guys have any "bag of tricks" for complex analysis problems, things that often come up as part of a proof technique. It seems most of the proofs draw on just a handful of few ideas, but that's just my experience so far. It would also be helpful if you guys can suggest any important theorems/topics I am missing. Any other general advice is very much welcomed!
EDIT:
To be precise, the main ideas/tricks/definitions/theorems I have compiled so far are:
- When is a function holomorphic
- Meromorphic functions
- Cauchy-Riemann Equations
- Liouville's Theorem
- Both Picard Theorems
- Analytic continuation (and its relation to zeros)
- Riemann Theorem on removable singularities
- Identity Theorem
- Maximum Modulus Principle
- Cauchy-Goursat Theorem
- Morera's Theorem
- Cauchy's Integral formula
- Cauchy's Residue Theorem
- Argument principle
- Rouche's Theorem
- Casorati-Weierstrass Theorem
- Residues
- Laurent Series
- The three kinds of singularities
- Useful theorems for calculating residues
- ML inequality
- Conformal maps (namely Mobius transformations)
I want to see what professionals and students have to say about their comparisons.
It could be based on difficulty, abstraction, time requirement, understanding, how much they have contributed to expand mathematics or anything you find appropriate for the comparison.
Given the rich connections between differential geometry and complex analysis is there a book that teaches complex analysis armed at say a graduate student who while comfortable with complex numbers and vector spaces has not really dived into complex analysis but has done a significant amount of differential topology and geometry?
Laplace with Neumann Boundaries, dirac-distribution, Einstein notation,... I have so hard troubles understanding all these topics. Im in my third semester of my bachelor in Europe and it feels like this is the hardest semester by now. Complex Analysis seems to cover so much more topics than the real one and four-momentum and co fuck my brain with notations.
This post is mainly a rant, but any tips on how to pass this? I would need an ELI5 of everything in complex analysis or a complex analysis for dummies. Same of everything Maxwell.
It this even possible or meaningful because I can’t find any?
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A link to the album with the corrections made: https://imgur.com/a/ha96a7R
Old link: https://imgur.com/a/VF2un9j
You can download the .pdf file and/or make any changes you wish to the .tex file from my Github repo: https://github.com/BhorisDhanjal/MathsRevisionCheatSheets
Hope someone finds this helpful!
I made this specific to my undergrad course so there might be some topics that you may have covered that aren't included in this sheet (e.g. Conformal maps).
I've tried to make sure there are no errors, but given the size there might be a few that slipped through. Let me know if you spot anything and I'll correct and update it.
Hello,
If anyone has previous semester resources for MAT 3121 complex analysis I'd be very interested in having them please :')
I would like to jump into some complex analysis. What all is needed to understand it? Also is there any good resources/ YouTube series to help learn it?
I have room for literally just one more math elective and I’m wondering if complex analysis would be a good choice. However, I don’t know if it has any applications outside of like physics.
I’m not sure how to go about this question. I tried using the definition of sinz in terms of the exponential and Liouville’s theorem, but I’m not able to get a final expression that contains |z|
Hi everyone!
Just doing some planning for 2022 and was wondering if anyone has experience with the Complex Analysis (MAST30021) lecturers? Very curious to hear any thoughts. For reference the handbook lists the lecturers as:
- Sem 1: Mario Kieburg
- Sem 2: Thomas Quella
Recommendations about which semester to do it in would be greatly appreciated! I am currently planning on doing it in Semester 1 :)
I’m finishing up my masters in math and heading into a PhD program. I’ve been enjoying my classes in (undergrad) topology, and grad complex analysis & measure theory. I’m looking forward to the next steps but I don’t know where to go from here!
What are some good courses? Areas of research?
Let z_1, z_2 be in D\{0} shot that |z1|=|z2| if and only if there is a unique biholomorphism from D\{0,z1} to D\{0,z2}.
D is the unit disc centered at zero.
I have room for one math elective and complex analysis sounds cool. However, I don’t know of a single application of it outside of like physics/engineering. So I’m wondering if there’s any point in taking it? Although, I have heard that it shows up in higher level numerical methods classes. Are there any applications of it in statistics? Will it help me anywhere in my PhD?
Hi everyone,
I recently gave some talks on an important complex analysis result -- The Schwarz Lemma. If anyone would like to check out this series, it is available here:
This murderer inspired the results of my Mathematics Ph.D. thesis -- Part 1 -- https://www.youtube.com/watch?v=AWqeIPMNhoA
The Schwarz--Pick Lemma -- The First Divide -- Part 2 -- https://www.youtube.com/watch?v=hd7-iio77kc&t=0s
Intrinsic Metrics on Complex Manifolds -- Part 3 -- https://www.youtube.com/watch?v=sHJ4LrhmzT0&t=0s
The Ahlfors--Schwarz Lemma -- The Second Divide -- Part 4 -- https://www.youtube.com/watch?v=ejs7SjNVpE0
The Schwarz Lemma in Kähler and non-Kähler Geometry -- Part 5 -- https://www.youtube.com/watch?v=-y5U-pwMQp8
This lecture series is based on some lectures I have given concerning a number of new theorems that I've proven recently.
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