What does a Complex Analysis PhD Qualifying Exam look like?

[@kailysrat118]

I haven't studied complex analysis in ~20 years but I may have some intuition on 5 for you. Basically if |f'(0)|

[@orpheus0108]

Im not in the same field but I always find it inspiring to see other people studying and imrpoving in their field. This stuff goes way over my head. Im okay at the math I stopped at (linear algebra, calc 2 and discrete math) but Physics was always so hard for me to grasp. It's definitly not a subject that is intuitive for me. I was so happy to finish university physics II and leave that in the past lol Ive got a couple exams/certs im studying for now and it can all feel so overwhelming at times. The amount of info and things to remember and look for. I find practing and building muscle memory is key. Im in cybersecurity. I am taking the Certified penetration tester specialist test next month and the OSCP a few months later. I feel ready, pwning dozens of medium-hard lab machines but Im scared ill get stuck or get real nervous come time for the test and screw it up haha. Anyways, thanks for the motivation and best of luck with grad school!!!

[@torquencol]

I don't have this level of math but your videos are really relaxing

[@BboyKeny]

Same, it's like listening to a foreign language but I feel like I'm getting more comfortable with it.

[@PhDVlog777]

Thank you 😊

[@50Steaks68]

Agreed. It's the voice, the dry colors, and the camera being held like a phone that does it for me.

[@JohnH777]

I recently took a Complex Analysis preliminary exam, and I have a Real Analysis preliminary exam coming up in August. I want to say that I'm in the same boat you are, and I have so much more to say about these types of exams, but I'm guessing you have it more than I do! All the best in your studies!

[@Adam-io6oz]

For 4 I don't think you need to use Taylor expansion. You can find a contour in D(0,2)\D(0,1) where g does not vanish. Then for large n |z^ng| will be larger on this contour than the maximum of |f| on it. Interestingly, this argument wouldn't work on the unit disc and in fact if f and g are both identically 1 then 1+z^n has no roots inside the unit disc. For 5, my instinct was the same as yours but I'm not sure how to argue via the Schwarz lemma. There is some a so that D(0,a) is inside Omega and which f maps into D(0,b) for some b. Thus, g(z) = b^(-1)f(z/a) is a map from the unit disc to itself (note that because we are working with discs to begin with we don't need the Riemann mapping theorem - i.e. the conformal map is just z mapsto z/a) and thus by the Schwarz lemma |g'(0)|=(ab)^(-1)|f'(0)|

[@Adam-io6oz]

By the way, we can avoid using Montel's theorem by using the Cauchy integral formula to write (f'(0))^n as an integral of f^(n)/z^2 which is bounded uniformly in n because f always maps into Omega.

[@PhDVlog777]

This comment section is the the real GOAT of YT lol!

[@__hannibaal__]

I was in complex analysis about 15 years ; i really enjoy that beautiful field of mathematic, in university it’s very ugly subject that i don’t like it ; so i do meine 1. The Generalized Gamma Function. 1.1 the Generalized Factorial Function : solution of equation, for initial analytic function f, found F() so : F(z+1) = f(z)F(z); 2. Complex Topology. 3. Modular Function, and analytical Number Theory, Zeta and related function 4. Integration over complex number, here i do too much. Also expansion in infinite product, and also so called general infinite summation of function and i not yet published. 5. Some research in logarithm properties and so 6. On solution of functional equation : g is analytic in D, f(f(z) = g(z); found f , dev, asymmetrical, limit ,,,,,,,…… 7. On Special Functions. 8. Using Mathematica … 9. Analytical Mechanics, Newtonian , Lagrangian, Hamiltonian. 10. Fluid Dynamics, Naviers-stocks equation, coupling model. 11. Analytic Solution of Elasticity Theory. 12. On some Physic Equations (PDE,… ) 13. Fractal study(not in deep)- Next-- 14. …… another staff … . Now i m in programming area , C/C++, some Graphisme , system programming , and some stuffs , … . Now i thinking to stop, and starting on writing my result, as Books and publications.

[@neerajnongmaithem392]

I know nothing about math, but i still buzz in from time to time. I just find it comfortable to hear a man share his life in such a honest way, i can feel the effort he puts in his study, the struggles he have. Its like hearing a close friend sharing about their life

[@EricaCalman]

In physics we have to take our own internal version of complex analysis but get to skip real entirely which was fine by me since if I never see another epsilon/delta in my life it'll be too soon. With the qualifier though that 'complex analysis' to me just means I can take a basic contour integral and use Cauchy's theorem to save myself some effort haha.

[@cemcengizyazc5701]

Exactly, I have just completed my Complx Analysis course (Physics Major) and all we did was to review limits, series then Cauchy and calculate some contour integrals with poles.

[@Aaron-lp3zt]

Pretty similar situation in electrical engineering as well. We have this very tailored course "Complex Analaysis for Electrical Engineers" at my uni that you'd probably think is a joke.

[@missstrangequark]

I'm also studying physics and in 2nd year we have two-semester class called Mathematical methods in physics but it is really detailed and hard, we use Butkov's book as an additional textbook.. but granted, we are allowed to have Bronstein with us while taking exams

[@karlinx]

i studied physics and got real analysis, now I got ptsd

[@briang.valentine4311]

2. A Green's function can be constructed that uses the symmetry of the domain. The singularity at the point (o,i) can be sent to infinity. Then the mapping is analytic and conformal.

[@yvonnesun641]

literally have no idea what's happening, but your voice is soooo soothing, so I play your videos as background calming music when I do my *basic* calculus 3 homework

[@michael-gary-scott]

the only maths i understood in this video was that 1am to 4:30am is 3 1/2 hours

[@furious5260]

i have no idea what is this but i find it enjoyable and relaxing lol

[@milesman1001]

I think for number 5 you can let F be the infinite iteration of f and show that f_n=f(f(…f)) uniformly converges to it on compact subsets of Omega. Then suppose by way of contradiction that |f’(0)|>1. Then, by chain rule, |f_n’(0)|=|f’(0)|^n, thus |F’(0)| is infinite, a contradiction to conformality. I’m a bit iffy on the first step but I think it should be fine.

[@Marryatau]

Thank you I’m still learning real analysis though I’m glad to see others do more experienced math I’ll keep at it and I hope you do too!

[@Mehdi-qp1tz]

Hi! You've done a great job documenting your work. Have you ever thought of digitalizing everything and making it all available online for people to use even decades from now?

[@Epilogue_04]

Im a physics undergrad working on me thesis about quantum computation but i have to say ive always had a massive respect to mathematicians, even i got in love with one mathematician girl 😂.

[@tererenji]

Hey! For Q5, I think you can use algebra! take the group Aut(omega) for maps f preserving the group identity 0, and given that It's a group under composition, 1. differentials also form a group, 2. Aut(Omega) is isomorphic to SL(C) or SL(Z) (SL for boundedness, Z if it's not connected)!

[@50Steaks68]

I’m nervous for your exam later this summer to be honest, like if it doesn’t work out what’s your plan? When you said the same guy was making half the test I got very worried for you

[@PhDVlog777]

I am becoming more comfortable with his types of questions. A new guy writes the other half, so I am less worried this time around.

[@MathsEtoile]

For 5, here is a proof using the 1st Montel theorem (every uniformally bounded sequence of holomorphic function admits a subsequence which converges uniformally on compact sets) fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_Montel You start by letting f_n = f^(°n) (nth iterate) which takes values in Omega. Using Montel theorem (which applies since Omega is bounded), you can extract so that it converges uniformally on compact sets to some function g. That is, (f iterate phi(n) times) converges uniformally on compact sets to g (where phi(n) : N -> N goes to infinity). Then, g'(0) = lim (f iterate phi(n) times)'(0) = lim |f'(0)|^{phi(n)}. Since the limit exists, this forces |f'(0)|

[@nazo4472]

I love how applying to a PhD is memorizing harder things than what you used to memorize for highschool.

[@nicob9279]

How many hours a day are you studying and are you doing it like every day? even when there are no exams? and how do you focus?

[@PhDVlog777]

I study at least a few hours everyday. I don’t pick a time, just topics I want to cover. As far as focusing goes, I am not sure I have a good answer. I just know that I won’t pass if I don’t study and that keeps me motivated.

[@nicob9279]

@@PhDVlog777 thats probably the right method :D ty for answering are you just using the cards u write to remember all the theorems? or are u using different methods, too?

[@marksd5650]

And I thought my calculus and differential equations finals didn’t make much sense. I looked through my son’s math books (engineering) the day before his graduation and there were more letters than numbers in his books. I have no idea what you’re talking about. I know that for my doctoral exams I studied a few weeks and then I didn’t want to look at it anymore. I was done.

[@cleverclover7]

That first question would bug me too. Personally I think the analytic universe is simply too large to not be explicit about what will be tested on, and questions like that favor the ability to memorize vs understand imo, though I'm not saying someone couldn't find the problem/lemma intuitive.. You're super smart and humble good job!

[@aaronstibeljr2584]

Please make a day in the life video

[@drewgonzales4533]

Cool stuff bro I look forward to pursuing pure mathematics and quantum physics..looks like fun.

[@o0QuAdSh0t0o]

On a scale of 1 to 10, where would you place your stress level?

[@PhDVlog777]

Today: 2 Day before the exam: 9

[@cloudz_wrld]

You mentioned that you have a way of memorising the content at 2:33, could you elaborate on it please?

[@John-ru4gz]

Can you make a video going over your organization techniques? How you save everything that you document ,and you pave out your schedule.

[@basitakram]

Seeing these many papers, how do you keep organized? Why don't you use digital notes?

[@godanipicprofile]

I am math major student, undergraduate student, and currently studying 3rd year and I don't understand how to even begin with when I look at exam questions in this video. Should I be worried? my math knowledge base feels very weak and more I study math, more I feel like this is not the path for me in uni.

[@PhDVlog777]

I don’t think you should be worried. These questions are not easy and are meant for students with masters degrees in mathematics. They become much more approachable after having read a few books on complex analysis and doing many problems.

[@giovi2348]

Sorry to bother you but i have a test soon, how can i say that a differential form is closed if and only if is exact, in R² minus axes?

[@giovi2348]

Sorry for my english, i didnt study math in englis so far

[@vortex5h0ck28]

I would ask r/learnmath or mathExchange if you want a quick answer

[@rohanghoshdastidar1047]

I also took my Linear Algebra + Numerical Anakysis + Complex Analysis exam this Monday ^_^

[@PhDVlog777]

I hope you aced it!

[@rohanghoshdastidar1047]

@@PhDVlog777 Nahh man I got a B 😭😭😭😭😭😭 But that's alright . . . . that was in 2nd semester . . . . In my 4th semester, I had a course on Laplace transform and Fourier transform and I got an A :))

[@padraighill4558]

I think for Q5 you could consider the fact that f is an endomorphism so there is some 'global' weak contraction property. Since f(0) = 0 points in B(ε) get mapped back into B(ε) so for any x,y in B(ε) | f(x) - f(y) |

[@prod.kashkari3075]

I’ve heard complex analysis is the “coolest” analysis. Is this true?

[@PhDVlog777]

It’s the easiest one imo, but I prefer measure theory.

[@tortillajoe]

For #3 I tried a partial fraction decomposition and got three separate integrals. I assumed such a γ exists and applied cauchys integral formula. One resulted in πi/2, the second in -πi/2, and the third in 0. To account for multiple loop or changes of orientation, it could be any integer multiple of those answers. But since no integer linear combination can result in 2πi/3, no such γ exists. This took me like 7 minutes so I might be missing something but this just what came to mind.

[@leonardolanciano3625]

Problem 5: WLOG you might assume f'(0) eq 0. If f'(0) = 0 we are done. Then, f is locally injective at zero, so f maps biyectively a disk centered at zero to another disk of radius r centered at zero with smaller radius r' (This is because f maps omega to omega) . Then g(z) = r/r' f(z) and g(z) = 0 and g(z) its a biholomorfism from the ball of radius r to the ball of radius r. Then you can prove by the riemanns mapping theorem that every biholomorphism that has a fixed point has that its derivative on the fixed point has module less than one. Hence, you get |g'(0)| \leq 1\Rightarrow |f'(0)| \leq r'/r \leq 1. Boundedness its not necessary.

[@alen2971]

What can you do with this knowledge?

[@98danielray]

​@@alen2971so you just go around asking meaningless questions?

[@minetheskybutter6881]

@@alen2971 Complex analysis is very useful in all fields of math, physics and engineering. For example propagation of acoustic waves which is relevant for the design of jet engines. Without it you probably wouldn't have the device that you wrote your daft comment on.

[@rogorix4991]

Maybe some fun exercices exist in X-Ens Orals exercices books

[@et2124]

Can you make a "tutorials" playlist, something like a playlist where you put videos where you actually go over the tasks(and solutions).

[@shiitarou933]

Ex 1 is wrong. The f_n need to be defined on a domain not on an open set. The name of the „lemma“ is called hurwitz theorem. It is easy to construct a counterexample if Omega is just open.

[@fibonacci112358steve]

Why did you put Arzela Ascoli in the "Complex Analysis" pile? Surely, it's from Real Analysis, right??

[@PhDVlog777]

It is a RA theorem but it was used in Stein Shakarchi’s complex analysis book to prove the Riemann Mapping theorem. That’s the only reason it appears on the CA list.

[@aleksandervadla9881]

How to Solve the second one?

[@matiastripaldi406]

you took the exam in january and it hasn't been graded yet? why do they take so long?

[@dziugaschvoinikov4440]

Do you really need to memorize a lot in math? Im studying math undergrad, and i dont need to memorize almost anything to pass the the bar. (I hate memorizing)

[@AerysBat]

Yes, you have to memorize. I studied math undergrad too, because I found it easy. Let me offer a suggestion so you don't make the same mistakes I have. Get used to doing hard work, putting in the hours, and pushing through things you don't always enjoy. Everything worthwhile in life is on the other side of discomfort, particularly when it comes to your career. Choose your goals and then do whatever it takes to achieve them.

[@lugia8888]

You memorize ideas and techniques to solve/prove other ideas.

[@PhDVlog777]

For some things yes. No matter the field of study, if you go deep enough, memorization is necessary. The more familiar you become with the subject, the less memorization you need.

[@lugia8888]

@@PhDVlog777 In one of Conway’s Complex Variables books (great resource to study for your Complex qual) he has a good quote regarding this. Something along the lines of “a good technique is worth a thousand theorems.”

[@frankchen4229]

yes; you will definitely need to memorize definitions and at least be familiar with popular theorems.

[@juliang8676]

So I am a PhD student in the UK, and looking at this I cant help but ask why do you have to do these tests? How does studying and practising exam technique aid in research and problem solving. Sure being familiar with material is great, but there is a huge difference between being able to remember and apply the general ideas and practises of proofs and actually being able to quickly recreate them. Seems like a waste of time to me but thats likely just a cultural difference, but something that one of the professors at my university (the head of the funding committee for phd students) said was that "good grades just show that you did your homework" and I cant help but agree with him. Also I may have misunderstood where you are in terms of your academic progression.

[@alimanski7941]

I was also puzzled, but I think it stems from how different graduate schools are in the US. They don't separate the MSc and PhD. So this replaces some sort of MSc final exam, maybe?

[@metalwing1232]

I do get somewhat concerned when you say “memorized”. whenever I was learning the more abstract subjects like abstract algebra/real analysis I really made sure I intuitively understood what I was doing because blindly memorizing certain problems isn’t the best strategy, math in my experience is you have a bunch of tools that you need to solve each problem with, it doesn’t matter if you memorized how to nail in one specific door, if you get a sliding door you need to nail together you won’t understand it cause you focused on memorizing how to do one problem. Since your a grad student I assume that you already knew this but I’ve watched a couple of your vids and they all have some degree “I memorized this problem” which might work for calculus but def not the analysis subjects

[@frankchen4229]

kind of; certain solutions and derivations might only work for very specific contexts so some recognition of a pattern from a prior similar problem within reason might be necessary.

[@mycrushisachicken]

your “understanding” is just memorizing the idea, meaning and applications of the subject. understanding is essentially just a certain way of memorization. and im pretty sure he implied that, since no where has he ever claimed to “blindy” memorize a subject. in fact, there is no scientific dissiplin where “blindy” memorizing anything, as you say, yields positive results in the long run.

[@Loots1]

Im taking complex analysis this semester ☺️

[@henriquepio2684]

looks like my graduation exam

[@themibo899]

I am looking forward to the correction of this

[@perriannesimkhovitch1127]

I am not a student: it's like not being married to a school. I am a Cal Poly Humboldt State graduate but LinkedIn practically relegated me to flunking the GED

[@ericstorm4839]

This inspires me

[@joebidens581]

Nice tip

[@swiffy2417]

Liouville, not Louisville… the i and ou don’t “commute”

[@injesusname3732]

higher math is just a hobby for people who like solving puzzles

[@joshuagasana2143]

Reading this comment section gives me a reason why I will never TOUCH this career path when I get to Uni☠

[@picesnl61]

Good luck with studying and on your test in two months

[@prathamesheedigi]

Brother , from which university are you ?? And when you are going to take on wall Street .

[@justahomeworkaccount]

Hello! Btw Why do you write in loose pages instead of a notebook??

[@xiaosheng2068]

My major is physics and I am just a freshman. I find it's interesting to learn mathematics, I want to study areas which relate to mathematics highly, such as General Relativity, Quantum Field Theory and String Theory. I have learned Advanced Mathematics and Linear Algebra. Apart from these, I learned a little about topology, group theory, ordinary differential equation and Riemann geometry by myself, just a little, you can even ignore them. So is there a list to learn mathematics, in order to study GR, QFT and ST by myself? Thank you so much!

[@PhDVlog777]

I am not familiar with much beyond the basics of physics so I cannot recommend much. What you’ve described sounds sufficient enough because beyond that is proof-based mathematics and I don’t see those courses being useful for you.

[@xiaosheng2068]

@@PhDVlog777 Thanks a lot! :)))

[@michaelnaretto3409]

To me, this may as well be in Chinese.

[@daeskk]

hi, where does a bachelor in mathematics work? im on a bachelor that allows me to follow many options after it, and Im considering math because I really liked calculus 1, 2 and ordinary differential equations but I didnt found anything telling me exactly what a bachelor in math do. can you help me?

[@PhDVlog777]

There are many options for those with a BS in mathematics. Teaching is the obvious choice but finance and banking are also popular fields.

[@daeskk]

@@PhDVlog777 thank you, im watching most of your videos and its encouraging me on following with math :)

[@ErikOnNoobTube]

@@daeskk You can also work with programming. Arguably the most expressive art-form, and just as fun as math when doing hard problems. I did bachelor's in mathematics into programming if you wondered.

[@frankchen4229]

computational and mathematical sciences

[@raghavdeshpande2905]

why do you use sheets of paper instead of notebooks ?

[@Jakkaribik1]

notebooks is at fault to not work anymore .. sheets of paper is later the test so might as well do it there Immediately

[@peterzeman2749]

What do you work on in your research?

[@Maxwell_Integral]

Pleasee bro post your notes online

[@solaris413]

which university u r IN?

[@maxpetrov4288]

is there any good books you could recommend for studying complex analysis?

[@TheMrKeksLp]

This is giving me so much anxiety. I'm so glad I'm done with all the math classes. No more math ever for me. I still sometimes wake up in a cold sweat thinking I have forgot to take some exam and will fail my degree

[@adavirus69]

I wonder where did you get so much problems from

[@ShaunYCheng]

Is this from United States?

[@sepehrjafari5223]

Hi your channel is great what is your field?

[@adrianzimmer973]

Am i the only one here that dosnt understand anything.

[@ilovereadingwuxiaxianxiaat8994]

Is this for all PhD's or just mathematic PhD's?

[@alimanski7941]

Mathematics...

[@trim1180]

Hey i wonder how do you connect paper sheets to a stack like that, do you use some kind of glue?

[@PhDVlog777]

Some are stapled but mostly loose papers. They are supposed to be scrap paper and just for me to practice writing up proofs on.

[@pablomingorance9378]

Gives my anxiety

[@devjyoti5614]

Bro how many phd s are you doing 😮

[@kirkpetersjr]

At my uni it's 4 questions

[@DerangedIntellectual9]

hello and welcome to another episode of "My Homework is Harder than Yours"

[@ihs2152]

A true mathematician uses A4 paper alone

[@patrickormerod3472]

I'm French I don't understand any of this

[@user-yi7kn7pt7h]

-i^2 st

[@victorgaray3012]

I have real analysis I next semester. Any advice? 😅

[@PhDVlog777]

Read Bartle and Sherbert if you are an undergrad. If you are going into grad school, then I would recommend Stein and Shakarchi. Kolmogorov is also a great book. Don’t be afraid to look stuff up on the internet. I do it all the time lol

[@victorgaray3012]

@@PhDVlog777 thank you sometimes when I look up things I feel like I failed. Thanks for the advice!

[@energy-tunes]

So how will this help me get laid exactly

[@Jakkaribik1]

Imagine you are Born to learn such Stuff Easy but never do it .. Yeah that's the Most Of the Population on every Subject

[@Jakkaribik1]

Education needs to change to give the chance to find people with such skills or others because as for now you must find such skills on your own in your freetime

[@Tux0xFF]

Whenever i have a difficult problem to solve, i watch Math videos and i realize: My problems could be 100x harder Or It could be worse Then I come back to my problems and they dont seem to be that difficult anymore 😅

[@lazarbaruch]

All nellow are of course just opinions. Your approach is quite ineffective. You should make a list of tricks, and you will find that there are not many. Next, see how they are applied in each case. Next, for each theorem, describe the plan of attack and pinpoint the gist of the proof. Most of the proofs in analysis, real or complex, involve several applications of the triangle inequality and other technical details. Please just skip trivial technicalities if you can remake them when needed. Try to get help from a colleague who can judge your understanding. Writing tons of pages of solutions will give you the feeling that you were okay, but the examiner was harsh. You are supposed to show research skills rather than problem-solving skills.

[@mikewei2619]

U r vary difficult mathematician....goog 1is those can make complex problem to snap

[@encefalord]

I struggle to understand how this PhD has almost the same value as a Medicine PhD where the guy does an disease incidence study...

[@MattHudsonAtx]

Yeah, it's not like math has ever helped humanity

[@nightmareintegral5593]

Formalist…

[@mathisnotforthefaintofheart]

BRRRRR

[@EsinaViwn9]

this is so useless yet hard.

[@kobemop]

Proof by contradiction is just so unsatisfying...

[@woflmao]

pretty sure you're speaking english but no f-ing clue what the heck you're saying 😂 not a mathematician btw

[@patrickormerod3472]

Shave some of that arm hair sasquatch

[@muhammadmustafa2946]

1st (posted 12 minutes ago)

[@icedthai]

Now go look at a Master's Thesis for Women's Studies.