What software are you using for writing and recording? I liked how you used different colors in your notes. I also appreciate how you post so much of your knowledge and understanding, publicly so others can attempt to look at and learn from your perspective. Interested in hearing you explain what is math? What is the purpose of? What is it about math is different from other similar quantitative academic fields?

The suggestion from class was to write a computer program, which is what I did. Rather than using properties of number theory to try to win down the possible fractions, I noticed that there are so few things to check, at least for a computer, it is better to just write a short program. I can write an inefficient program in just a few lines much faster than trying to prove results!

Valuable and very enjoyable lectures. Wish I had a Prof. like you in my Masters. But one question to the respected Professor. At 34:52 you ask to the students " What do you think I did?" .. but their answer is not clearly audible and even when you at 34:55 say "yeah ...", it still is not clear what property of number theory you are talking about and the subtitle often gives misheard words. So would you kindly like to say what property are you talking about to solve the problem? Thank you very much.

Will there be any change for anti-periodic function?

Thank you for making this public

This was perfect. I think this is exactly how maths should be taught. It is easy to imagine that Weierstrass was following a similar train of thought when he formulated his results in the 19th century. Thanks you for this lecture.

12:18. The infinite product does not converge if the input value z is not in the set of roots of f(z). This makes sense but is it intuitive? This strikes me as an exceedingly subtle point, which may have something to do with the “nature” of infinity. Just as there are infinitely many primes, and infinitely many integers, but the set of primes is still a subset of the integers, so it is here - there are infinitely many input values of f(z), and infinitely many roots of f(z), and yet the roots of the function is a subset of its domain. Is this a sensible observation or am I making this too complicated?

1:25! Not even 2 minutes in and this is already beyond excellent! If you say e^x we will make fun of you and embarrass you in public. 😂😂

Thanks, seems like a great course. I do wish this course used Ahlfors since it’s such a classic textbook, but Stein and Shakarchi is nice as well.

Sir, with an honour please make a playlist for this Group Theory Lecture series.

Comment from the web: gicoba72 challenge about the problem at 27:30 about 11^(n+1)+12^(2n-1) = 133 kₙ : can you express the generic kₙ in concise form, meaning not as a sum of n terms, but a simple expression?

How do you get to think about function like that 11:35, cause it seemed so intuitive how you made an example and described the behaviour of that function even though you are not looking at one

I really wish this video had more comments. I appreciate the enthusiasm. But math can feel so lonely.

sorry, cannot fix - can you try looking at the transcript? can also go to the homepage and look at the lectures from a previous year.

There are audio problems after around 40min

(Steven, you may want to rename the title of this video to Math383Fa23 instead of Math383Fa2023, so that it matches the other ones. Thanks for sharing. Cheers!)

Sir, create playlist for courses so that it can become easy to see courses

Hello you thank you ❤

Excelent thanks

I don't know if you still teach this course but I have a nicer, geometric proof that holomorphic, injective functions cannot have a zero derivative anywhere. It uses the concept of the winding number: basically take the formula to count the difference between the number of zeros and poles of a meromorphic function (the argument principle, page 90 of Stein & Shakarchi). You can prove this counts the number of times the image of the curve over which you integrate goes around the origin. So take a closed curve gamma. Then the integral 1/2pi*i * intagral_gamma f'(z)/f(z) dz = Z(f), with Z(f) the number of zeros of f inside gamma. This is also equal to the number of times f(gamma) goes around the origin. This is also easily seen in the following way. Move from the domain of f to the range of f. How many times does the curve f(gamma) go around the origin? This is then integral_f(gamma) 1/w dw = integral_gamma f'(z)/f(z)dz by the chain rule. To move from counting the number of time f(gamma) goes around the origin to the number of times it goes around a point w_0=f(z_0) we count the number of zeros of f-w_0 so it is integral_gamma f'(z)/(f(z)-w_0)dz since the derivative of w_0 is zero. Now, to prove the proposition, assume f holomorphic and injective but there is a z_0 where f'(z_0)=0. Since f' is holomorphic too, its zero's are isolated (f is analytic & not locally zero), there is a disk D centered at z_0 such that f' is only zero at z_0 and non-zero everywhere else in D. Okay, call the boundary of D the circle C and call f(z_0)=w_0. Consider now the curve f(C) around w_0. Since f'(z_0)=0, in the power series expansion of f centered at z_0 in D you can easily see that for all z in D: f(z) = (z-z_0)^k*G(z) with G(z_0) not zero and G holomorphic in D, for some k >= 2. This means that the multiplicity of the zero z_0 of f has multiplicity k (at least 2). So f(C) goes at least twice (k times) around w_0. Now you can pick a point w close enough to w_0 such that it is still inside f(C). Then f(C) goes around w also k times. Therefore there are k points (possibly counts with multiplicity) inside the disk D for which f(z)=w. These k (at least two) points are distinct since we took D small enough so that f' is only zero at z_0. So from the power series around any of the k w's you can see it has multiplicity one, since f'(w) is not zero! This contradicts the injectivity of f, which proves the result! This is a wordy proof, but if the groundwork of the argument principle in terms of winding numbers and counting the number of times the image of a curve goes around f(z_0)=w_0 is done, and it is established that if f maps z to w a number k times, f(C) goes around w exactly k times, this proof is very easy when you draw a picture with it!

What is the prerequisites for this course?

Glad you enjoyed.

That was such a Nice, Mind-blowing & Motivational intro 👌 Let's see what happens on later sessions ...

There was a comment here but I removed it as it seemed to be about monetarizing sites. Please only post items related to the math of the videos here; these are freely provided for the benefit of anyone as part of the pay it forward philosophy.

Your book probability lifesaver I just find mistakes in 53 page We call q the sample space or outcome space and the elements of q the events. elements should be subset

At 4:23 you misspelled Riemann as "Rieman".

What books would you recommend to someone who aims to learn univariate and multivariate analysis in a comprehensive theoretical and proven way?

39:45 thank you for going through this. By showing the "first instinct" to a problem, it feels like your redirecting a thought process. very intersing

I took Real Analysis in college... now I get to take Complex Analysis at home :) Approaching along the imaginary axis is so weird!!! I love it!

thank you - the entire class is freely available online

sir,salute to you for dedication and consistency .. constantly posting a video on youtube

Response to comment on audio issues: Sorry - I have switched to different technology. I know KZbin can often add transcripts…. Might not be able to for this

please 😭 ive been studying complex analysis and analytic number theory for the past 6 months, and only today did i understand why the zeta function is important, thank youu

At the start of this class, the question is asked if the heuristic calculation done on the blackboard is enough to prove that g is holomorphic. I would think the answer is "no". In order to do the calculation, the chain rule is needed. This assumes that g is holomorphic at z. So all the calculation tells us is: if g is holomorphic at z then f' cannot be zero at z.

This is brilliant!

There is no audio after 41:00

I have a question for you by the proof of lemma 1. Why is it true that the length of BC is greater than or equal to 2? thanks a lot!

Hello sir , I want to ask you something. I have come across different versions of cauchy theorem like homologous form& homotopy form of cauchy theorem. But want to understand the homologous version of cauchy theorem and integral formula. But I am kind of confused about these two versions. Are these two versions (homologous &homotopic versions) different? Could you please suggest me some materials/books about these topics ? Thank you

I start number theory this semester in January. You’re so awesome, I love your teaching style and could only hope for a professor like you. You’re encouraging of thought and i feel like I could ask any question without feeling insecure. Thank you ❤

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Gale Shapely's theory is misused in todays world. It doesn't typically account for people with disabilities.

You!! You are back on KZbin.

you are a king btw

It breaks my heart to be forced to choose between so many amazing topics to learn. I am deeply fascinated by coming to a true mastery of complex numbers, functions as vector spaces, and how all of this ties into my knowledge of quantum mechanics as a linear theory. If only I could stop time and learn all of math while the universe waits for me!

hey, i have a question. in 39:56, we have f(z)=1-e^iz / z^2, and we are taking the real part of it. my question is, why is this valid? shouldn't the z^2 term in the denominator affect Re(f(z))?

the integration of (1-cosx)/x^2 was fun, i seem to like this course? thank you!

i had so many 'aha' moments in this lecture, i love this!

I like how you discuss the context and develop the ideas before writing theorems and proofs. I think that is one important difference compared to just reading the book.

Really nice lecture, thank you.