Bro, I'm just a 14 year old boy from central Europe and I'm very grateful for these youtube tutorials, nice work I think this video deserves to have more views than it really has. I have a question, where do you animate these videos, is it just python or some special program, I ask because I want to do some math videos also. Thanks for the answer.
This video is just amazing! It explains analytical continuation like no one has ever been able to do so. Not only that, it explains the nature of complex numbers and functions especially the illusive logarithm which i had a great difficulty understanding when i took complex analysis in the university! The explanation is so deep that after the 50 min presentation one is able to retrace the whole argument of analytic continuation from really scratch in ones own mind! I sometimes wonder why such easy concepts were taught to us in the university in such a convoluted and incomprehensible way!!! And I also wonder how much time was put to produce this beautiful video. The fact is that many people will appreciate this effort tremendously!!
I. POSTED A COMMENT ON THE PREVIOUS VIDEO. I'M A RETIRED EE, AND I HAVE BACKGROUND IN BASIC CALCULUS 1,2,3, DIFF EQU'S, SOME VECTOR , & TENSOR CALCULUS, SOME LINEAR ALGEBRA, SOME FUNCTIONS OF A COMPLEX VARIABLE. I'M WEAK ON THE FIELD THE PROBABILITY SINCE I NEVER TOOK A COURSE OR READ A BOOK ON PROBABLY AND STATISTICS. I'M JUST FAIR IN MATHEMATICS. I'M FAIRLY GOOD WITH MOST CONCEPTS. SO I DO WATCH AND CAN UNDERSTAND FOR THE MOST PART OF THESE MATH. IDEAS POSTED ONLINE. BUT I STRUGGLE AT TIMES. BUT YOU SIR, ----- YOU ARE A MATHEMATICS GENIUS. AND THESE LAST TWO VIDEOS I WATCHED ARE ABSOLUTELY AMAZING. I WAS NOT AWARE OF THESE CONCEPTS. I REPEAT THESE VIDEOS ARE ABSOLUTELY AMAZING. I AM ABSOLUTELY DUMBFOUNDED IN THE LAST TWO VIDEOS. I'M GOING TO CHECK OUT MANY MORE OF YOUR VIDEOS. YOU ARE A SUPER TEACHER. YOU TEACH MATHEMATICS VERY CLEARLY SIR. KEEP UP THE AMAZING WORK LET ME DO ON THESE VIDEOS. YOU HAVE BENEFITED MANY MANY THOUSANDS OR MORE SO MUCH WHAT'S THE IN-DEPTH DETAILED EXPLANATIONS YOU PUT FORTH IN THESE VIDEOS. YOU TEACH WITH EXTREME CARE AND CONCISENESS IN YOUR DEFINITIONS. IT IS SO WONDERFUL TO FIND A REALLY REALLY GOOD MATH TEACHER, REALLY KNOWS HIS STUFF. THANK YOU MUCH FOR THESE VIDEOS THAT TEACH SO MUCH ON ARE SO INFORMATIVE ON MATHEMATICS PRINCIPLES.
On 35:50, basically everything must be symmetrical by induction. even cases are like a U shape always and odd cases are like a ^u kind of up down up shape, and symmetry carries over to the next case when you integrate, because the second half and first half of the interval alternates between being the mirror image horizontally and the mirror image both horizontally and vertically of the second half of the interval. By symmetry and the fact that the start and end values of the polynomials at 0 and 1 are the same, if said value is not 0 then the odd cases will not have an integral of 0 as the left and right half will not have areas that cancel.
Hey, I just recently discovered you channel. Is there a possibility that you would make a video about automorphic forms? It seems that you mainly focus on analytic number theory. So it may serve as a rich and interesting topic to discuss on your channel in a more informal fashion.
In fact, the first series cannot be equal to 1 because for any finite k, there is always a residual term of -1/(k+1). If you mean that the limit of the sum as k->∞ = 1 then OK but I don't think your logic is correct because no matter what infinity is, when you evaluate the sum, you have to use an actual number and then you have the residual - hence the limit.
_"when you evaluate the sum, you have to use an actual number"_ The partial sums form a sequence, and this sequence converges to a limit. So what are you talking about? The sum of a series is usually defined as the limit of the sequence of the partial sums.
As was mentioned by other people here, the middle term at 46:45 is wrong (lacks a multiplicative p^-s term). So the sum at the right starts at k=1. This is correctly done in the following. Such minor errors could be mentioned in an "errata note" in the introductory text, or pinned at the start of the comment section.
when will the next video come out? looking forward but my complex analysis exam is tomorrow eek, your vids have really helped so far! (especially with the intuition)
40:58 Natural log of -1 =pi since 180 degrees converted into radiants equal pi if you choose to go opposite way to -1 ofcourse you get -180 degrees or - pi Why? Because complex numbers dont have imaginary part ,they have rotations ,and true form of rotations is in radiants
This is a wonderful video with very clear and intuitive explanantions of stuff and pulling in someone who has detatched for math a long time ! Thanks a lot for all the effort in this content creation.
Excellent quality, it has been a while since a bingwatched hours of math videos. Thank you for motivating me to pick a complex analysis book again. Please tell me you are releasing another video!
It is a pain for me not to donate, but gosh these videos are wonderful. I've never seen a video so well animated anywhere. The rhythm is perfect, the explanations are clear and are not just dumbed down examples. Some quick proofs to really convince are shown The amount of time and effort put into this clearly pays up for the waiting time Man this is perfect, I just killed my evening watching all at once
Hi, I was wondering how you proved the odd Bernoulli numbers vanish? I've found proofs online but they rely on an alternate Taylor series definition of the numbers. I also saw a proof in the comments that claimed that odd index implied odd degree, which in turn implied that Bk(x) was an odd function and therefore must vanish at zero, but not all odd degree polynomials are odd functions (e.g. (x+1)^3 is neither odd nor even). Please let me know what you did!
These Bernoulli polynomials (see 32:45) would be either odd or even functions if you shift them by 1/2 to the left. So perhaps it would be more straight forward to define them on the interval [-1/2, 1/2]. However, I suppose that this would result in clumsy formulas in the applications. So it was decided to define them on [0, 1], at the cost of slightly more complicated coefficients.
Is it just me pre is this video deeply disturbing. I get the feeling that something is wrong in our sense of math that this is trying to explain. Such that math and it Beauty stems from something we don’t yet know.
Wonderful video. Finally an exposition on the zeta function that goes beyond merely saying "the zeros of the zeta function tell us something about prime numbers", but actually demonstrates it.
As an engineer who just studied complex calculus as a ‘process’ to solve problems in book to pass exams, this video is truly enlightening. I m just a hobbyist now with no real goal to apply it in real life but the satisfaction I got after watching this video is amazing. Pls make more of these. It’s been a while since the last video was posted.
