What do complex functions look like? | Essence of complex analysis #4
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- Опубліковано 3 лип 2024
- A compilation of plots of different complex functions, like adding and multiplying complex constants, exponentiation, the power function (including nth roots), and logarithm. Issues like branch cuts, branch points, and branches in general will also be discussed as the result of inability to construct the plots. Finally, we will do a 4D rotation (composed of two 4D reflections) to the typical Riemann surfaces pictures, and see that it should be the same as its inverse functions.
The video is going to be jam-packed with visuals and animations, so while it may sometimes be too quick, you can pause the video; or you can just simply appreciate the visuals, the plots, and the animations.
Some interesting plots are usually the vector plots, like for the power functions, we have different regions of flow. The formula of 2(n+1) when n is positive can be left as an exercise - it is not TOO difficult to see why, but it is not the focus of the video, or not the primary feature that I want to discuss; or for negative powers, we have dipole, quadrupole, and octupole, and in general multipole, which might be familiar to physicists, because in electromagnetism, we use multipole expansions to see the dominant effects of the electric field.
Watch the previous video to see what the 5 methods of visualisation I am referring to, and also watch the Problem of Apollonius video for the next video!
Video chapters:
00:00 Introduction
01:01 Adding constant
02:51 Multiplying constant
06:14 Exponentiation
09:47 Power function - integer powers
14:11 Power function - complex inversion
15:39 Power function - square root branches
20:37 Power function - Riemann surfaces
22:53 Logarithm
26:51 Logarithm - 4D rotation
Music used:
Recollections - Asher Fulero
Stinson - Reed Mathis
Beseeched - Asher Fulero
White River - Aakash Gandhi
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If you want to know more interesting Mathematics, stay tuned for the next video!
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As always, pause the video if necessary.
I realise that making a whole video series about complex analysis is a really monumental task - much, much, much more than what I expected - but don't worry, I will still make them *eventually*, just that (1) I need a lot more time, so the next video is not going to appear at least until mid / late Dec, and (2) I might have to sometimes switch up, i.e. only occasionally putting out videos on CA, and not necessarily all uploads would be about CA. This is a problem I found myself into, like the group theory series, that I feel like some people might not like CA all the time (or at least, will get bored after something like half a year), which is why I want to give a heads up that this might happen if I feel like it.
This video is much better viewed with a good degree of familiarity of the stuff mentioned at the start, so PLEASE watch those first; and of course, watch the video on Problem of Apollonius as well for the next video on Möbius maps.
Nice I might buy a mug man ... Glad you've got a monetary support structure external to YT internal issues... Also added a comment about electrical engineering and how we use the Möbius transform as a tool via the smith chart
This time you outdid yourself, an absolutely incredible job! It gonna be on the must-watch list for my students. You set the bar really high this time, you inspire me to put more work into my videos!
Thank you so much for the kind words!
This is why I freaking love complex analysis
LOVE complex analysis. These visual tactics to SHOW complex functions is pretty cool. Thanks!
Glad you like them!
sedenionic analysis is better
These animations are brilliant! This really makes clear what a branch point is and why it's necessary for sqrt and log. I can't wait to see the visualizations for the Mobius transformation!
Thanks so much for the appreciation!
My favorite are the Re-Im plots even if they are less useful because looking at them I am in awe of the beauty and the complexity of the 4th dimensional world they came from. In the others it's not quite so obvious to see.
The inverse function on the z-w plane is really nice and helpfull, thank's !
It's clear how much effort goes into these videos, thank you! Great work!
Thanks so much for the appreciation!
What serendipity to have your next video be on moebius maps just when I will need it for uni - brilliant series man.
Hope that it will help!
Just discovered the channel today. I am a big fan of how you explain the topics in all your videos. I look forward to seeing what you make in the future :)
Thanks so much for the kind words!
Those 29 minutes went by fast. I can't believe you managed to get that much into one video. Looking forward to the Möbius maps video!
Thanks so much!
Wonderful video! Little gems like this are one of the things that make me keep loving math 🥰
P.S - Try not to worry too much about your French pronunciation. My French teacher (who is French herself) once told me that "French words are like Christmas trees: full of seemingly ornamental letters".
Oh thank you!
Lol, that's what other language speakers say about English :D
@@Neme112 I've seen a native English speaker say that about English...
Thank you so much for this series. really amazing
Awesome channel and video!! Im glad somebody is finally making videos to illuminate the beauty of complex functions
Glad that you enjoyed it!
This actually solves one of the mysteries behind a fractal that I found. I didn't know why there was a discontinuity along the real axis and now I know, because I'm taking roots of the inputs. I also managed to find a single white pixel at the origin which would be the branch point. The exact roots also depends on the inputs which demystified what frankly should've been obvious which was white around the point -1. The specific roots that I was taking was z+1. When z is -1, taking the reciprocal before using it as an exponent makes it blow up to infinity. Now I'm curious what I'd get if I tried plotting it in 3D so it would avoid the branch cut.
Worth the wait
Thanks for the appreciation!
wonderful video... currently I'm reading about translation, rotation, inversion, magnification and this video helps me a lot to understand z and w plane visually
Many thanks :)
Branch cut? More like “This is the stuff!” I’m enjoying your videos immensely and I can’t wait to watch the rest of them.
wow! amazing video! the apollonius video was the first of your videos i watched and i absolutely loved it! thanks for making this content. waiting for the next video :)
Glad you enjoyed it!
🤐 and 😭
How could any complex analysis course go on without these kind of visualization!! Thaaaaaaaaaasank you
BTW this 28 min video needs 2 weeks to internalise.
These visualisations do take a fair bit of effort, which is probably why a lot of courses wouldn't do these.
Can’t wait for the next video about Möbius Transformations!
Enchanting and illuminating
Thanks!
I like the z-w plane method the most, but the 3d plot showing different 4d perspectives is really amazing and mind-boggling when I first saw it!
I love vectorial representation and its links with magnetism
Super cool, keep on
Can't wait for the mobius map video
Brilliancy as your standard..so we thank you and get messssmerised by your presentation...
Thank you so much!
Wow made me feel nostalgic of complex analysis.
Merci pour cette vidéo.
As always that's a beautifull video
Thank you very much!
Thank you so much for this explanation! I understand theta much better now. ^.^
The video actually came! Never doubted of it though. Amazing :)
Thanks so much for waiting!
I feel like the "this picture comes from f(z)=1/z" was made for me 😔👉👈
Thank you!!!
nice explanation .. really apprecitable
Thanks!
oooh can’t wait for midterms to end so I can watch this!! 😂🥳
Brilliant and Beautiful!
Glad that you enjoyed it
Amazing video. Reveal such bizarre information for the one, who studies complex analysis, that you're like "AAAAAAAAAA"
I'm really glad I found your channel when I did ∫
daaaamn this is haard work
Thanks for the appreciation!
asking wich way to plot I like most is like tell me to choose an icecream flavor
very nice
Really comfortable hearing this guy explaining in clear-spoken English, supongo que no es 👳🏾♂️
tnx!
Waiting for next video
Got an idea for a t-ahirt: At 17:16 there's the image of the branch cut, really nice pastel colors, it would be really cool that image on a t-shirt and the colors extending all the way through the shirt including the back (though the back coud be in a pinkish white, instead of white)
That exponential 3d Plot be looking dummy thick
First, Cant wait to learn about the concept!
Hey, I think there's a small mistake at 5:03. Matrix multiplication on the right side of the equation, the second row first column spot should be "ad + bc"!
Yeah, I was wondering about the same. A second check is that complex number a+bi is represented by a matrix [[a, -b],[b, a]].
Sweet.
Thanks
Excellent job! Can I ask you how you generated the flowing Polya fields?
Looks like UA-cam algorithm is withdrawing from drugs
Great videos, they help me a lot. Do you have a tutorial on how to plot the figures you present (software, code, etc.)?
Thanks for the appreciation. For software related questions, you can read the description.
13:20 The quadrupole is normally arranged in a square.
Yep, that's right. The reason I did that was essentially making it clear that it is just composed of two opposite dipoles, and also that it aligns (somewhat) with the vector field generated.
i like the 3d plot
I love Mod-Arg plots. Argument just maps so well to hue that other graphs just confuse me :)
I think z^(1/2) color plane has spin 1/2 so the branch cut is an effort to project the helix in a 2d plane
Around 5'55'', what is the function you use to transform z into e^z smoothly? When you were transforming z into z+c the intermediate steps were clearly z+tc with t going from 0 to 1. To display cz, you used z->tz, with t going from 1 to `c`. However, I can't guess which methods generated the intermediary step when you switched progressively from the identity function to the expoential function
Do you have any recommendations for best available graphing software for complex analysis? I'm a fain of the level curve transformations you use in this video for example.
Amazing visuals!!!
But my only feedback is that you should explain the dynamics and should add some rigor before showing your visualizations, that makes your visualizations much more beautiful than before.
and this is what makes visual complex analysis book beautiful.
for example, while showing the z-w plane graph of exponentiation you should have shown that e^x corresponds to magnitude and e^iy corresponds to arg, so, as x increases magnitude increases and as y increases arg increases
Thanks! I hope that was more clarifying when I showed and explained more in the domain coloring plot, but thanks for the feedback!
i created a method of visualisation of complex functions which uses movement through a time axis as one dimension. shal f:C->C ,z->f(z)=r(z)*exp(i*phi(z)) be a complex function in its polar form with z=x+iy. Then consider a parametric Plot g_t:R^2->R^3 , t element [0, 2pi] as folows:
g_1(x,y)=x
g_2(x,y)=y
g_3(x,y)=Re(f(z))*cos(phi(t))+Im(f(z))*sin(phi(t))
if you plug in any complex function and 3D-Plot with t as parameter which you can control afterwords, you can spin around in a cycle through the 4D Graph of the function displaying one angle at a time. if you would animate it, so that t continusly goes from 0 to 2pi in a loop, you get a "4D" animation, which represents the whole function. Also you could role around other Planes, like the input Plane to get more insight.
I found this method very usefull, when i studied complex functions in more detail.
In regards to polya vector fields, what is the exact definition of them for a beginner? Is it just an animated vector field, and what does the animation depict? I am not too sure, sorry if I have missed anything :)
@11:30, can someone explain why the identity function divides the plane into 4 flowing quadrants? I would have thought it would just be a static vector field since each vector is just itself after identity?
13:46 well it also makes sense since two sets of two charges are grouped up, so they just act as a stronger version of a monopole, so it can also be viewed as just the 6 charges
Thought about this as well, but the thing is you can use the same argument to a quadrupole (there would be 3 monopoles together in this case), yet there are 4 loops for a quadrupole...
@@mathemaniac hmm yeah that is weird. then the way id think of it is: the octapole is a combination of two "opposite" quadripoles side by side, their opposite nature means that a plus an a minus meat in the middle and "combine" the sections that the quadripoles would generate by themselves. This doesnt happen from the dipole to the quadripole since their equal charges meet in the middle. i'd think this holds for higher numbers aswell. but then again it doesn hold for monopole to dipole, since those dont "combine" in the way i've described.
@@mathemaniac great visuals btw
I love it. 😁👍
Am I correct in my observation that the flow of the vector fields in the integer power parts look hyperbolic? 🤔
Thanks!
Yes, it IS hyperbolic. You can even prove this is hyperbolic, if you know a very useful tool in fluid dynamics: streamfunctions.
@@mathemaniac Cool 😎
At 4:58 in the multiplication section, isn't the bottom left element supposed to be bc+ad as opposed to ac+bd?
Why are you not using black for zero in your domain colouring plots?
any way to get the code to generate the riemann surfaces?? I tried a lot but could not finish it.........
@4:55 there's a mistake in the multiplication of matrix.
How the vector plot is generated? What software is used ?
I just lerped the input and output points and it looked almost the same as the exponentiation one
Complex sin cos tan and hyperbolic counterparts please.
Apollonius Circle
Why does the octupole only have 6 loops? Seems it should have 8? Would the plus minus bubble things be arranged in a ring alternating plus minus? I'm not an expert in this subject matter, just trying to learn and understand. I'm probably not understanding something.
Hello author, I've recently been working on a mathematical physics chronicles content and I very much need to edit 5 seconds of footage using the footage from the video you created, I really, really need the 5 seconds of material. I'll be sure to credit the source of the footage and @ your channel when I'm done with my work. I very much recognize your work and thanks again. : )
(The above text was translated by me, using a translation program, so please forgive me if I use the wrong words)😀
If it is just 5 seconds of footage, and you credit the channel, feel free to use it.
bro beautiful
Inversion transformation
Fun fact: In power functions, the number of branches corresponds to the denominator of the exponent
@Shimmy Shai *reduced fraction
Can anyone explain why the identity function vector field is split in 4 parts while the z^-1 function has the shape of an magnetic monopole. For me they look swapped, where is my mistake?
I was plotting the Polya vector field, so I am plotting the vector field generated by the **conjugate** of that function. This convention is explained in the previous video on how to visualize complex functions in general.
The advantage of using Polya vector field is that it makes complex integration a lot easier to visualize, and also that it actually represents invsicid, incompressible, irrotational flows.
@@mathemaniac ah yes, this makes sense of course. Thank you for the fast response.
And for the fantastic video, I love it!
This video slaps ass. Incredible animations great explanations
Thank you!
0:05
Puisque peut cree un ordre dans la desordre , exemple les nombre premier le probleme la difernte p(n+1)-p(n) la desordre malgre π(x)=1-k +x/π +1/π ×la somme de 1 linfini de sin nπx/m /n p(n+1) - p(n) =2m k appartiene 0 vers l infini la correction donne que l unicite dans la circonference [nπ/2, (n+1)π/2] de p l axe de sin .
Couldn't you visualize the branch points with helices? i.e. just visualize it as a multi-valued function extending in both directions. (Or as a single-valued function returning equivalence classes; same thing.) (You could also highlight a particular argument equivalence class on the helix with a different color, to indicate the "portal" the helix is periodically passing through.)
Are the 3D plots of square root and logarithm in the video what you mean?
69k views. Nice ;)
22:00
نحتاج لترجمة please
25:10
Howdyou get the thumbnail
11:18
dream
23:34
"since we chose the black color for 1"
I think you meant 0, you kept saying 1 instead of 0
I did intend to say 1, because I meant that the color at 1 should be black, because log 1 = 0.
@@mathemaniac ahh okay, awesome video btw, I really enjoyed it
Nice visuals, thank you, eg, those transitions between inverse functions.
Why not use "true 4D" visuals, is what I'm always wondering. A "simple" tool like Graphig Calculator 4.0 offers it, so why not more powerful tools?
Compare for instance the Log z item (together with its inverse Exp z) with my "4D" video on it at
ua-cam.com/video/YUmMXPE3Mpo/v-deo.html
and many more functions on my "4D" channel there.
Sometimes I fear humanity get stupider each generation. Now I don' t as much.
I hardly understand ur vedio.
Did you watch the previous ones on the playlist?
Also, the way you talk about the "hue" is very non-concrete. A mathematician's way of describing it is "The modulus is cyclic, and cycles n times for every 2pi radians along the unit circle.". But, you're not a mathematician, quite the opposite actually little buddy. 😊
It's more traditional to label the second angle φ, and the second radius ρ. But okay, whatever makes you feel better little buddy. You know nothing about mathematics.
18 ad spots? ! Bye
The images go by too briefly. It's better to linger on them a while without audio. For someone who already has a handle on complex analysis the talking is the least interesting part.
What a boring introduction...
I have viewed your entire playlist of complex analysis. I have learned much and thank you very very much.
After watching your videos, I went exploring a list of complex analysi at Wikipedia. Reading is a pleasure.
en.m.wikipedia.org/wiki/List_of_complex_analysis_topics
Yes, complex analysis is a beautiful topic! The playlist is far from complete, since each video takes a long time.
Mathemaniac I understand the lengthy time of creating animation and video editing. Do you make your own videos? What softwares do you use? I am guessing you use Blender, a 3D modelling software.
I make all my videos, but for software, please read the description.