Contour Integrals

I had to edit out an example because I stopped before I finished and claimed my half answer was the full answer 🙃

[@dilation1057]

Complex analysis is a blessing and real analysis is a nightmare.

[@someguypbwhw8]

Indeed

[@motherflerkentannhauser8152]

For those inexperienced in this matter, here's why: 1. There are many convenient results in complex analysis that you cannot use in real analysis. For example, you'd have to worry about different classes of differentiability of functions on the real line but if a function is 1-time complex differentiable, it's not only smooth but also analytic automatically. There are also powerful tools not available to real functions such as residue theorem, argument principle, cauchy integral formula, etc. that can help you evaluate integrals (even real, improper integrals) without anti-differentiating. 2. The many annoying technical details that you have to worry about in real analysis should still exist in complex analysis if you want to nitpick or when you get advanced enough. But I supposed in many elementary courses they don't talk about them because you're usually assumed to have learned how to deal with them in real analysis. Sometimes I even suspect they know complex analysis is harder and therefore, deliberately tuned down the difficulty. So complex analysis courses and exams "appeared" easier than real analysis.

[@dilation1057]

@@motherflerkentannhauser8152 I always though Complex analysis was harder, but the sheer amount of ways complex analysis can be used makes it easier to understand since You can approach it from a more “physics” perspective rather than a pure math perspective. Although I have never studied Real analysis much (simply because i am lazy) the way you can visualise complex analysis helps a lot in studying it.

[@jacques8277]

You're actually a fantastic youtuber! I love your work and the fact that you're so humble and down to earth. Good luck with your studies and keepdoing great work!

[@PhDVlog777]

Thank you for the kind words :)

[@rohanghoshdastidar1047]

Idk why but I LOVE this channel, I stumbled upon it on the HOME page and although these topics are unrelated to what I have now in my course, this automatically has increased my interest and enthusiasm towards studying maths! 😁😁 Edit: I LOVE your handwriting!!

[@PhDVlog777]

Thank you haha I've had mixed reviews over my handwriting. My students don't like it, but my fellow grad students DO like it. IDK lol

[@rohanghoshdastidar1047]

@@PhDVlog777 Your students have too high standards 😂

[@aravindakannank.s.]

@@PhDVlog777 then they would like to suffer from cramped eyes in case of seeing all your viewers handwritings

[@David-pq6wt]

Enjoy the channel. I am a mechanical engineer 10 years out from my BS degree. Rarely use anything more than algebra and trig at my job but I self study calculus regularly bc I enjoy doing problems. Your videos are over my head bc as an engineer we take calc 1-3, diffeq, LA, and a math methods course that focuses on partial diffeq. So I have no analysis knowledge or upper level algebra. But my friend has a Phd in math and we we’re in under grad together. I was so impressed when he would tell me about his Real Analysis course and how demanding it was. He struggled but did really well and it was one of his more satisfying classes that he ever took he told me. Since then I have always had an interest and respect for RA. Possibly someday I will study an introductory text on it. Good luck on your exam whenever you take it. Hope the studying goes well and keep the good videos coming

[@FndfnnFnefnfn]

Wow, I like when some student's got the idea and like to share their way of understanding. Thank you very much!

[@larryyonce]

Good examples, and I like your style . Ignore the trolls (there are a few). You have great content and many of us are interested in your PhD journey 👍 Have a wonderful day.

[@PhDVlog777]

If you don't have a few trolls, then you are doing something wrong >:)

[@daniellindner826]

This was amazing !!! I want to pursue a math degree in the future and complex analysis always seemed so interesting to me. Thank you for this video. I'm looking forward for more content from you.

[@TechnoCoderz369]

Broooo Idk how to thank you! I'm 14 and I really wanted to learn contour integration cause my friends know it. They make fun of me and they say I cant learn it as Im dumb. Thanks bro you explained it very clearly! Thanks brooo! Love from India

[@nevaehjol9319]

I wish everyone on the earth explain things like you. Thank u

[@ayrthhhn]

Mathematics is the queen of the sciences, and Complex Analysis is the queen of mathematics

[@doscoroll2366]

It would be great if you made a video casually explaining what analytic, holomorphic & entire functions are! I guess I'm not the only one that doesn't really understand the mathematical language used in textbook explanations etc.

[@emit_etinif]

Dude, watch your language! Pursuing higher levels in maths takes passion and a TON of work. Let me remind you that you’re doing really well, and that your continued hard work sets you apart and is inspiring to others! You are doing really well. I’ve watched serval of your videos and I noticed that you’re criminally hard on yourself. I can’t imagine you’d use the same language towards somebody else who’s studying maths. And there’s always some genius in the department, who breezes through the material, and you can’t help but to harshly compare your own progress to that person’s, but remember you’re that genius to somebody else. Be kinder towards yourself, you’ve earned it! :)

[@PhDVlog777]

Thank you, I know I criticize my own work heavily, but I think it comes with the territory. Mathematicians seem never satisfied with their work, but we still publish and it is appreciated.

[@vandel_]

This is literally what I am doing right now EDIT: Please make a video about the residue theorem and how to solve hard integrals with it

[@budiardjo6610]

the way you learning is so much fun

[@abdulllllahhh]

Theres a time in ones life before they learn complex analysis and contour integrals, and a time after they learn them. I truly believe they are some of the most beautiful things to come out of this subject, and I’m kinda sad only math majors cover this.

[@RagibMahirAshhab-sn8qv]

well guys if u think that this is easy he might be ur 1st contour itegration teacher.

[@darkpikachu_.]

plz push more content on functional analysis and measure theory🙏😊

[@RichardJohnson_dydx]

I am so glad I stuck with engineering. I love math but if it's not applied math, I am so clueless. The most I've ever done with complex numbers is using the Laplace transform. Even then we had a table and had to do some algebra to find time functions. ODE's and control systems weren't so bad.

[@David-pq6wt]

I enjoyed control's too. I took Continuous Control Systems and aside from a brief intro at the end of my DE class that is where I learned the most about Laplace transformations.

[@biswaranjan17]

I don't know but I love this channel so much I'm also a student of mathematics in UG and I'm very very interested in philosophy of mathematics, so how could I begin with to get into this field of study (Philosophy of Mathematics)... Please any suggestion as such would help me a lot...

[@azimuth4850]

Very nice primer on this

[@jabra1115]

What do you think of the Analysis 1,2 books by Terence Tao

[@motherflerkentannhauser8152]

The hardest I have encountered is where it is unclear which branch cut or branch point to choose and as a result, I couldn't even decide which contour to draw.

[@PhDVlog777]

I struggle with the same thing, that pesky complex-valued logarithm...

[@jiachengzhao3834]

You should be a professor or at least a lecturer. Love your videos.

[@cerret9703]

I’m pretty sure z^z is analytic throughout the complex plane no?

[@Alidaher2468]

This was really helpful i hope that you make a playlist about contour integrals and residue theorem🙏

[@rescyy2235]

Hello there, I wanted to ask, do you have some recommended PDF files of textbooks from which I could potentially study all that you study, not necessarily just complex analysis from this video, but multiple areas of mathematics? Thank you

[@PhDVlog777]

If you want to study Real Analysis, the Royden Fitzpatrick book is my favorite, his pdf document is free online. For complex analysis, I think the ahlfors book is really nice and is also available online for free :)

[@rescyy2235]

@@PhDVlog777 Thank you so much for the quick reply!!

[@_mario_2985]

What do you think about the importance of studying at a "well-known" / "famous" university? Do you think it really matters when you want to work in the industry (finances)? Im asking because I will finish my undergraduate degree (mathematics) at my (hometown) university soon, and I'm still trying to figure out what I want to do afterward. I would like to leave my comfort zone and study statistics at another university, but I don't thing I could get into a "famous" university, and I would still need to do some English certificates before I could apply.

[@abebuckingham8198]

The advantage of prestigious universities isn't the courses, it's the contacts you make there. Wharton has a great reputation both because smart people attend but also because rich people attend. It allows the ownership class to mingle with the professional class in a mutually beneficial way. The value of status is cultural as well with some cultural caring more than others. For example name recognition of the university your attend matters a lot more in China than it does in the US.

[@mastershooter64]

networking is like half of what makes you succeed in acedamia, the other half is passion and hard work

[@PhDVlog777]

The absolute most important thing is who you know, and what kind of connections they can give you. The prestigious universities do have more connections, but that is not to say the smaller, less well known universities can't get you a job equal to or greater than that of someplace like Harvard.

[@_mario_2985]

​@@PhDVlog777 That's what I am often told. However, I'm still not quite sure what is meant by giving connections? Do/did you actually participate in some programmes where you get/got to meet people from the industry? Do/did you talk to your professors so that they could be a reference for your future applications to jobs? Because here in my undergrad programme I did not really get/take the opportunity to make contacts but maybe this is because I didn't try hard enough or does this come at a later stage of the studies?

[@Kitty_Kankles]

Is the term “disk” interchangeable with circle here? Like a disk with infinitesimally small thickness?? Also, is green’s theorem related to the Cauchy-Gousat theorem? I feel im getting flashbacks to calc iii here

[@navierstokes2356]

No advanced integrals by residues? :(

[@mastershooter64]

residue theorem in C^n

[@Parzival659]

what's the name of the pen ur writing with?

[@harmonicfunction]

thats a sharpie ultrafine marker!

[@dailymasala614]

Hi sir, please upload more videos. Love you sir

[@TechnoCoderz369]

Please make a video on Lebesgue integration pleaseeee

[@abebuckingham8198]

I call C the circle numbers, not the complex numbers. Calculating the eigenvalues of a quarter turn rotation matrix convinced me they're fundamentally geometric and I often define them geometrically for my students. Circles are far less intimidating than the algebra makes them feel when you approach them from that direction.

[@pursuitofnumbers]

What's the pen you are writing with?

[@Pure_Imagination_728]

Complex analysis looks delightful compared to real analysis.

[@PhDVlog777]

Most of my fellow grad students like Complex more than Real. I think they had had an undergrad course in complex before the grad level course, but I never did. The graduate level complex analysis course was my first exposure to it and I was intimidated by it. By comparison, I have had real analysis a few times, and was used to it by the time I got the the graduate level course. So for those reasons, I prefer real, but I get what you mean. Complex is much easier to get a grip on.

[@siamsama2581]

This is so cool! Make more videos! I'm doing Theoretical Physics and I have some basic complex intro to complex analysis this Friday (lecture). I think I study more of it next year though. But yeahh upload more! Plus the relaxed tone of the video is really nice and I like the casualness of just a pen and paper and you're handwriting is neat. I'm in 2nd year btw. Today I had some group theory test and my god it was so hard, seems like the lecture notes didn't help and I prepared for it but nahh Idk what I'm doing it. Even the class questions seem so hard and I feel like the lecture notes and the Qs are unrelated for the vast majority of the questions. But anyway cool video and make more please!

[@siamsama2581]

And the unrelatedness makes it so hard coz Idk what I'm doing or where to start

[@PhDVlog777]

I will cover more on complex analysis and the difficult stuff in the near future. Sorry to hear about the group theory test, I was never strong in that subject to begin with. Ring Theory and Field Theory were more my speed. Thank you for watching :)

[@oraz.]

Isn't the first example integral non elementary?

[@cluelessMuslims-ed5js]

I found a new hyperfixation lol

[@sachalucienmoserferreira2233]

nice video

[@tfg601]

it's exactly u-substition.

[@kaka63211]

bruh this is easy as shit, we learn this at school...

[@Tennisbord]

No one asked what you learned in school

[@High_Priest_Jonko]

26 minute long video from Struggling Grad Student? You best believe there's going to be a bunch of RAMBLING. Just get to the point, brother. No wonder you're struggling so hard!

[@jacksonyukihillary]

shut up

[@High_Priest_Jonko]

@@jacksonyukihillary So you're saying you disagree that this video could be edited down a little?

[@jacksonyukihillary]

@@High_Priest_Jonko yes, I am

[@planthub9252]

Love your vids!