The 5 ways to visualize complex functions | Essence of complex analysis #3

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Complex functions are 4-dimensional: its input and output are complex numbers, and so represented in 2 dimensions each, so how do we visualize complex functions if we are living in a 3D world? There are actually 5 different ways to visualize a complex function, and this video is going to explore a bit about each of them.
Some of you commented that you have already studied complex analysis in full, but hopefully there are still some things that you haven't seen before, because a typical university course on complex analysis wouldn't contain as many visuals as seen in this video.
I know this might not be recommended by RU-vid as much simply because the video is not that long (less than 20 minutes), and it seems like RU-vid only puts my videos in recommendations when my video is very long. Originally I wanted to put things that will be covered in the next video into this particular video, but I figured that it doesn't make sense to cram two quite separate things into one video just for the sake of watch time. So please consider sharing this video, liking and commenting so that more people can watch it!
LEARN A LITTLE BIT MORE HERE:
More about complex visualisation: www.nucalc.com/ComplexFunction...
Joukowsky transform: complex-analysis.com/content/...
Joukowsky transform (NASA): www.grc.nasa.gov/www/k-12/air...
Credits to Yehuda, there is an interactive tool to obtain the domain colouring plot for complex functions here: people.math.osu.edu/fowler.29...
Music used:
Heavenly - Aakash Gandhi
from RU-vid audio library
Video chapters:
00:00 Introduction
01:03 Domain colouring
03:35 3D plots
05:45 Vector fields
07:50 z-w planes
10:53 Riemann spheres
Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:
forms.gle/QJ29hocF9uQAyZyH6
If you want to know more interesting Mathematics, stay tuned for the next video!
SUBSCRIBE and see you in the next video!
If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!
Social media:
Facebook: / mathemaniacyt
Instagram: / _mathemaniac_
Twitter: / mathemaniacyt
Patreon: / mathemaniac (support if you want to and can afford to!)
For my contact email, check my About page on a PC.
See you next time!

Опубликовано:

7 авг 2024

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Комментарии : 399

@LookingGlassUniverse 2 года назад

Beautiful ❤️

@mathemaniac 2 года назад

Thank you so much Mithuna!

@vinestreet4031 10 месяцев назад

Unfortunately they have many more distractions too.

@lorenpearson1230 Год назад

So many advantages for young people learning now. We were once constrained to 2d paper and plotting the planes and piecing them together. Lots of mental gymnastics to intuit where they were interesting and choosing where to spend more time and the manual calculations. These visual representations are game changing for modern math and makes you appreciate what Euler did by hand.

@mathemaniac 2 года назад

This is one of my proudest works (aside from Jacobian one) so far, but since it is relatively short compared with my recent videos, it might not get into recommendations, so if you enjoy this video, please like, comment and share this video! Of course, if you want to and can afford to, please support this channel on Patreon: www.patreon.com/mathemaniac I would say this explicitly in the next video, but if you want to get a head start, please watch my video on Problem of Apollonius, because it is relevant to the discussion of Möbius maps.

@aphleesegurtra2820 2 года назад

I wouldn't have explained it with the maths, However, the graphical illustration allows for visual representation to formulate a cognitive understanding of relationship between the nodes. For the image presented-fact.

@jacobhoward7579 2 года назад

Awww shit! You gonna cover my boi Möbius!?

@mathemaniac 2 года назад

@@jacobhoward7579 Of course! But in a future video.

@Bestofchatgpt 2 года назад

Check this out. Alot of people say trippy but it's so complex its not understandable my a none math person. ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-uH5CFPUXT9U.html

@PrafulGagrani 2 года назад

The difference is the double root at 2+i and single roots at +/-1 of the complex height function plotted. Visually we see the winding number is different, particularly each hue appears twice at the root 2+i.

@mathemaniac 2 года назад

Yes - that's the answer! However, although I know what you mean by "winding number", it might not be the most appropriate word here (?), since it could really mean different things, but yep, your observation is right!

@polyhistorphilomath 2 года назад

The winding number when viewing the map as composed with addition of a phasor of small modulus (tracing out f(2+i+εe^i(θt+φ)) ). Something like that.

@mathemaniac 2 года назад

@@polyhistorphilomath Ok, that makes more sense, since winding number usually refers to the number of times the *curve* has wound around a point, and since there is no curve here, I was not too sure whether winding number is an appropriate term here.

@Xingchen_Yan 2 года назад

Is this the same (or similar) of f(x) = x^2 having two roots both being "0"?

@Djake3tooth 2 года назад

@@Xingchen_Yan yes, it's a squared term so you'll see this in the plot as a double winding of the colors

@AbrahamLozadaabe 2 года назад

In my opinion, vector fields are the best. At the end, the complex numbers are the vector space RxR with a product that makes it an algebraic field.

@gonzalodiaz2752 2 года назад

I studied engineering and with this video, complex analysis and complex algebra finally made sense to me.

@mathemaniac 2 года назад

Glad to help!

@gonzalodiaz2752 2 года назад

@@mathemaniac Thank you for this marvelous work ❤️

@sinecurve9999 2 года назад

3:17 The zero has multiplicity 2. The hue goes through two full cycles as you go around it.

@mathemaniac 2 года назад

Yep - that's true!

@maxwellsequation4887 2 года назад

Genius!

@squarerootofpi 2 года назад

Thanks. I suspected the function isn't injective around that point, because of the square, but couldn't put my finger on the word.

@quadrannilator 2 года назад

Extremely lucid explanation. And a new light into visual representations of complex numbers. The aerofoil mapping example was mind-blowing. Please keep these coming. Superb animations. Thanks!

@mathemaniac 2 года назад

Thank you so much!

@yash1152 2 года назад

yeah, the aerofoil mapping example was really great

@MedicenMulix 2 года назад

My favorite is the z-w planes. Since I'm a huge fan of hyperbolic geometry and how Möbius transformations acts on the complex plane, it is very useful to think on complex functions as transformations of the plane

@pacolibre5411 2 года назад

For the mod-arg plots, one modification that I’ve seen that I like it to use a logarithmic scale for the output. This works because the modulus is never negative, and it makes all of the zeros look like “negative poles.”

@mathemaniac 2 года назад

Yes - that's also an option!

@matyaspoko 2 года назад

"This looks like-" A PRINGLE!! "an airfoil." oh... Wonderful video and explanation as always!

@mathemaniac 2 года назад

Haha, now that you pointed it out, it does look a little like a pringle :)

@yugecheng8941 2 года назад

Exactly my reaction

@RedStinger_0 2 года назад

To me, domain coloring seems like the most versatile option, but a z-w plane in the context of subsets is probably my favorite.

@johnchessant3012 2 года назад

Amazing work! For the z-w plane, I hadn't thought about how there are lots of different ways to animate it before, that was really cool!

@mathemaniac 2 года назад

Glad you like the video!

@PunmasterSTP 2 года назад

Ways to visualize? More like “Facts explained beautifully before my eyes!” This is an incredible series and I can’t wait to watch more.

@inverse_of_zero 2 года назад

I like the first (colour) mapping the best. It is the most natural, neutral, and easy to visualise :)

@user-gs1fb2et2y 8 месяцев назад

my favorite way to visualize the complex function is vector field. It establish a bound to the multivariable calculus and give me deep insight of complex function. Besides, I love your video sooo much, it indeed help me a lot in perceiving the complex function!

@SimchaWaldman 2 года назад

My favorite complex plottings in descending order: 1. Domain colouring. 2. Mod-Arg plot. 3. Re-Im plot. 4. Streamplot / Pólya Vector Field.

@GoogleUser-ee8ro Год назад

This series of complex analysis is gold mine. You revolutionized the way of teaching high dimensional abstract maths.

@agustinbrusco7173 2 года назад

this is as beautiful as it is complete and honest with some of the limitations of each method. as I spent my free time the last few months making an interactive complex function visualizer in python, which includes domain coloring, plane transformations and 3D-surfaces (Abs-Arg, Re-Arg and Im-Arg where my choices), I really appreciate the work behind each of these explanations and examples. congratulations on this amazing work, greetings from Argentina

@mathemaniac 2 года назад

Yes, each plot has its own pros and cons, which is why we want different methods to fill up some shortcomings of the others. Thank you for your appreciation!

@crustyoldfart 2 года назад

I think you are justified in feeling proud of your work presented here. What you are tackling is a HUGE field, which you are approaching apparently from the POV of a mathematician. The field of what I learned to call ' conformal mapping ' is an important technique in the ' applied ' fields and engineering. The most common method when I was growing up was the w,z technique when computers and fancy graphic techniques lay far in the future. The mapping had to be done point by point and graphed by hand. In my private life I am a retired [ mechanical ] engineer, amateur photographer, painter and sculptor, builder of model aeroplanes and designer of model racing yachts. So I am inexorably drawn to your work. Personally I have found that with the arrival of computer graphics that 3D plots are the most useful. I use MAPLE and their plot3d( [x(p,q), y(p,q), z(p,q) ],p=p1..p2,q=q1..q2) ; being a fine working tool for rendering 3D surfaces in various styles. The same function can be used to construct line segments in 3D which are likewise very useful for visualization, especially since the form so generated can be manipulated dynamically to explore the form from any angle. As a comment on the use of colour, the photographer part of my brain kicks in to be aware that the brain manipulates our perception of colour, so that in a sense the colour we think we see is overwhelmingly influenced by its surrounding colours. My takeaway from this is that use of colour as an indicator of parametric value must be used with care.

@mathemaniac 2 года назад

Thanks so much for the appreciation!

@crustyoldfart 2 года назад

If I may be permitted to make an additional comment : The representation of numerical data by means of graphics is actually a topic dating back at least to 1854 during the Crimean War. Here Nurse Florence Nightingale is credited with the invention the pie chart. Also, one recalls that geographers and cartographers have been dealing with this problem for a long time. They use colour routinely. Perhaps their most successful technique is the contour map, which of course is a simple way of presenting a 2D rendering of a 3D surface.

@NexusEight Год назад

I like your hobbies. Then, when I went to give your reply a thumbs up, I discovered that we have the same first name! So from one Harold to another, good job.

@crustyoldfart Год назад

@@NexusEight Yay ! Harolds of the world unite ! Thanks for the appreciative remarks.

@Bestofchatgpt 2 года назад

Great video. As I teach myself mathematics I find videos like this very important to help me create the art I create. You might enjoy them if you like complex numbers

@benYaakov 2 года назад

I loved the vectors representation . Even though I am new and couldn't catch all , but glad to watch . Nice work

@pepesworld2995 2 года назад

wow this is brilliant. thanks a lot. i got a bit lost on the last 2 but im sure i'll get there later. i find i rewatch videos like this every few months. i understand more every time :) i have added this to my permanent rotation.

@cansukuyumcu8231 3 месяца назад

As a physics enthusiast, I find the vector field method to be incredibly powerful and effective. Thank you for providing such insightful video; it has truly enriched my understanding.

@nrp_g 2 года назад

Great video! I'm excited to learn a lot more about complex functions than the little I know already. Awesome animations and clear descriptions.

@mathemaniac 2 года назад

Thanks! Glad you liked it!

@harshitkhandelwal3251 2 года назад

You made me change my major. Probably best decision I have ever made. Thanks

@mathemaniac 2 года назад

Wow! Haven't thought that I could have such an impact for such a small channel!

@harshitkhandelwal3251 2 года назад

@@mathemaniac ya, I was majoring in economics and minoring in maths Started watching 3blue1brown, mathologer and your videos Got hell of interested in maths And now I am majoring in maths and minoring in econ Lol.

@jimdeligiannakis6314 2 года назад

Definitely prefer the vector field the most. You dont even skip any variables in your visualization! Although maybe my fluid mech background means im biased... Great video!

@yash1152 2 года назад

yeah, the point of not skipping a point is definitely awesome. buttttt, yeah, one negative side could be that visualisation from this is essentially partwise, i.e. some loss of detail is there as well.

@__-op4qm 2 года назад

@@yash1152 zoom in for detail?

@HAGARCIA 2 года назад

Função de Complexo em Complexo. 5 representações gráficas! Espetacular!

@brian8507 2 года назад

I think the sphere is the best.... because it shows the duel nature of zeros and poles at infinity.... and it shows that e^z has an essential singularity at infinity (just like e^(1/z) has one at zero) It also shows that there is only one point at infinty....

@yash1152 2 года назад

> _shows that there is only one point at infinty...._ nice. yeah, +inf and -inf are coinciding that is? 😅

@flov74 2 года назад

@@yash1152 Yes they are on complex analysis and enabling this enables you to actually define a way for dividing by zero without too many paradoxes.

@yash1152 2 года назад

@@flov74 i have also thought about that many times when thinking about the plot of 1/x. if +inf and -inf coincides for x, you get a nice curve resembling the shape of tan x. and thus discontinuous only at x = 0.

@yash1152 2 года назад

even that discontinuity at x=0 is resolved if these are taken to coincide for y too. but i couldnt visualise/imagine what that shape would look like 😅 . That is, couldnt figure out how to think about 2 axes being simultaneously folded.

@ethandole2218 2 года назад

I like vector fields the most because they're really pretty! :D but all of these are super helpful tools for visualizing complex functions

@mathemaniac 2 года назад

Yes - all of them are very useful tools of visualisation!

@dominiquecolin4716 2 года назад

@@mathemaniac it is both great and the visualization make sense to the human brain. Unfortunately, it is both discrete and normalized: I understood that all modules are set to one. did I get it right ?

@IshaaqNewton 2 года назад

I won't regret subscribing this channel. Amazing work, man.

@mathemaniac 2 года назад

Thanks for the appreciation and the subscription!

@322luisao 2 года назад

Grande vídeo, grandes animações e grande conhecimento por trás delas.

@Pranav-vb5ho 2 года назад

Great work! Came across your channel for the first time, loved it very much! Keep up the good work and keep the videos coming! :)

@mathemaniac 2 года назад

Thanks for your appreciation!

@thomaswatts6517 2 года назад

Love love love your content, I'm amazed at how prolific you are!

@mathemaniac 2 года назад

Glad you enjoy it!

@maestroeragon 2 года назад

Subbed and Belled! Great content about something i’ve been recently very interested about, can’t wait to learn more about complex functions

@mathemaniac 2 года назад

Thanks for your appreciation!

@antoine2571 Год назад

Amazing! Congratulations !! This is such a massive amount of work

@saraswatasensarma6041 2 года назад

I am a first year Mathematics student from India. Thank you so much for making my day brighter!!! Your videos are works are works of art, beautifying the concepts which on first exposure often appear dull and mysterious.

@mathemaniac 2 года назад

Glad you like them!

@brricasushi1033 2 года назад

The two full circles of hue are indications of a change of exp^{i*2\theta} in the function output around the second order root from a change of exp^{i*\theta} in its input. So a 2\pi full round result in a 4\pi hue. This video is making so much sense!!! Thanks for visualizing those bored classes!!!

@nathanisbored 2 года назад

3:25 its symmetric around that point and i bet its cuz of the multiplicity of that zero in the top (the one thats squared on the outside) i guess i like domain coloring the most just cuz it felt very intuitive after you explained how it worked, and also the zeros are obvious. also 3D is just hard for me and the other 2D ones are too sparse.

@mathemaniac 2 года назад

Nice observation, although "symmetric" isn't exactly what I intended - but yes, it has something to do with the multiplicity! What is another difference between these two types of points?

@rektwatermelon6746 2 года назад

Vector field approach is my fav...it gives us a 'natural' way to define integral of complex functions and many things from vector calculus can b applied....although Riemann sphere is ingenious in its own ways

@angeldude101 2 года назад

I know of another way of representing complex numbers that treats them separately from vectors, but also lets you multiply them. The vector field plot works perfectly for this since you can literally represent the complex numbers as transformations of vectors.

@xxxuselesspricksxxx1481 Год назад

this video made my jaw drop, especially with the joukowsky transform, for me who's studying aircraft engineering it surely was fascinating

@adarshkishore6666 2 года назад

Another beautiful video! It must take a lot of hard work and time to make these amazing animations, but viewers like me love every bit of it! Plus the explanation was excellent, and it intrigues me to go deeper into these topics (no one really bothers to explain domain coloring so much :/ but you explained it in a manner that was great!). Hope to see more videos coming in the future :)

@adarshkishore6666 2 года назад

P.S.: I love the z-w plane method the most, I think it is the most obvious for seeing the conformal mapping.

@mathemaniac 2 года назад

There will be more videos! And yes, thanks for appreciating my efforts in making this video!

@xinlinli459 Год назад

Great animations and detailed explanations! Extremely helpful! Thank you!

@mathemaniac Год назад

Glad it was helpful!

@lina31415 11 месяцев назад

great video! i like all these methods, but the 3d plot and vector field methods are probably my favs. also, you have very androgynous voice, it's quite nice to listen to :)

@fluffymassacre2918 2 года назад

Man you completely outdid yourself with this one keep it up

@mathemaniac 2 года назад

Thanks for the kind words!

@curtpiazza1688 Год назад

Wow!! Powerful stuff!

@AJ-et3vf 2 года назад

Awesome video! Thank you!

@tim-701cca 10 месяцев назад

That’s great😊 Never know some ways before, learn it now❤

@angel-ig 2 года назад

Nice video! I really appreciate the great animations you've created.

@mathemaniac 2 года назад

Glad you like them!

@MZaki-db7ll Год назад

Complex variables very complicated methods to solve complex problem. As an Engineer and Scientist always solved such problem and Numerical numbers were collected using different Numerical methods software solver. As well as a Graph can give how total systems changing within Limits which is very important to Optimize a system or design a product. Thank you very much for such clarifications and different ways to explain everything.

@sodiboo 2 года назад

I don't know which one i like most, but i can imagine that if you love the color wheel and angle/magnitude representation of vectors, the domain coloring might be the most beautiful, because it does map precisely those to colors in a way that does make complete sense. Personally, i don't really like either of those things and i find it hard to read, but that might just be because we are visualizing 4D points onto a *2D* canvas, and it's just hard to understand 4D, not this visualization specifically

@powerdriller4124 2 года назад

Overlapping the two 3D plots of functions: f( Xr, Xi ) --> Zr y g( Xr, Xi) --> Zi , that is: The Re-Im and the Im-Re plots, we detect the Zeros when the surfaces of both functions intersect together with the Z=0 plane (aka complex plane).

@Kaepsele337 2 года назад

I like the z-w plot, but depending on how it's used it might be important to label the grid lines or give them different colors. Domain coloring also has its uses, but that's just using the z-w plot for the inverse function and use the color wheel instead of the grid ;)

@chandansaha4311 2 года назад

Thank you. Please continue the series.

@mathemaniac 2 года назад

Thanks for your appreciation!

@cycklist 2 года назад

Wonderfully explained! My compliments. Also thank you for spelling colour correctly ❤️

@mathemaniac 2 года назад

Thanks for the compliment!

@bangaloremathematicalinsti5351 2 года назад

Great video really enjoyed...An artistic and aesthetic demonstration

@mathemaniac 2 года назад

Thanks for the appreciation!

@anywallsocket Год назад

Love how the airfoil shape is seen to be a particularly angled view of the pringle outline shape, thanks to your animation 😮

@jimzhu6315 Год назад

Thanks for making this video! I learned something new from it. Just wanted to point out that the Riemamn sphere should be sitting on top of the complex plane with its south pole located at the origin 0+i0. Then, draw a line from its north pole through the sphere to any complex number, and that number will be mapped onto the sphere at the interception of the line with the sphere. This way, there is a bijective mapping of every complex number between the plane and the sphere, with concentric circles in the plane mapped to the latitudes on the sphere, and lines emitting from the origin in the plane to the latitudes on the sphere. In particular, the unit circle in the plane is mapped to the equator, and the infinity circle in the plane to the north pole. This is particularly useful in dealing with the infinity of complex numbers. In many ways, it is like the flat-Earth vs spherical Earth.

@awildstevey 2 года назад

I took complex analysis a few years ago and we did not cover domain coloring. That’s a very nice and intuitive idea. Very cool.

@missoulasam 2 года назад

Great Video! Your depth and animation, WoW!

@mathemaniac 2 года назад

Thanks for the appreciation!

@hisxmark 2 года назад

The Riemann Sphere corresponds nicely to cosmology. If an n-dimensional sphere is projected onto an m-dimensional cartesian manifold where m

@AlessandroZir 2 года назад

very nice video!! thank you; ❤️❤️🤸

@nano10067 2 года назад

Great video, this helped me a lot to understand my applied mathematics class!!!

@mathemaniac 2 года назад

Glad it helped!

@clementdato6328 2 года назад

I used to first visualize complex functions with mod-arg plots as my first choice for the 4th dimension is always the colors, let alone here when it comes to the arg which topologically corresponds well with the hue. It seems however from the video we see little merits of this method. Maybe I should try more others.

@mathemaniac 2 года назад

That is the most natural thing to do, and actually it is very useful if we were to think about singularities or zeros (Mod-arg plots), but there are other methods too, which complements the things that we couldn't see when we restrict ourselves to just the 3D plots.

@2false637 2 года назад

Really enjoyed this!

@mathemaniac 2 года назад

Glad to hear that!

@lekonda5526 2 года назад

Nice! Thank you for your efforts!

@mathemaniac 2 года назад

Thanks for your appreciation!

@arongil 2 года назад

Great video!

@mathemaniac 2 года назад

Thanks!

@Diaming787 2 года назад

I like the 3D plots with the 3rd axis being the magnitude and the angle being the color. Helps you visualize how "big" a complex number is. "Big" meaning it's magnitude.

@PrashantKumar-zc5pj 2 года назад

Can and gonna binge watch the full series. A request‐ please do a video about tensors.

@skilz8098 2 года назад

I don't particularly have a favourite. I'd say it depends on the context in which you are trying to portray or demonstrate your I\O along with the problem domain. It would be similar to choosing a programming language... it would depend on the project or current problem that you are working with. It would come down to choosing the appropriate tool for the job at hand as they each have their pros and cons or strengths and weaknesses.

@joy2000cyber 2 года назад

Wearing a VR headset, ZW planes can be easily displayed, z-plane on the floor, w-plane on the ceiling, a line connecting input and output from floor to ceiling, that will clearly show the property of the function.

@jclaer 2 года назад

Thank you!

@silentsnake-qd7xi 6 месяцев назад

I’m currently watching this in my college math class. Nice job!!!!❤

@plwn6468 2 года назад

Vector fields do feel more homely. However each method is useful for some purposes.

@Mr_Mundee 5 месяцев назад

my favorite is the mod-arg 3d plot, it contains the most information and in a very understandable manner

@samirelzein1095 Год назад

out of the 5 methods i like you most!

@usamahasan6378 2 года назад

Amazing video, I simply like the z, w plane over reimann because the simplicity of transformation between.

@marius3023 2 года назад

Very good video bro. Keep on grinding ❤🔥

@mathemaniac 2 года назад

Thanks for your appreciation!

@earthscrust9092 2 года назад

Awesome, please make this a series that contain everything in the book "Visual Complex Analysis" plus do applications of each key concept plus make those ideas your own and give a really really indepth intuition of key ideas, plus God help you because there will be great effort from your part because the book is just a guideline at that point. And thank you and the people that are just like you. May you live well.

@mathemaniac 2 года назад

Thanks! As stated in the introduction video, I have already stated this book forms the basis of some discussions in this video series, so don't worry about that.

@fordtimelord8673 Год назад

I have that book “Visual Complex Analysis”. My favorite book on the subject.

@erictao8396 2 года назад

This is such a cool video!

@mathemaniac 2 года назад

Thanks!

@mahxylim7983 2 года назад

I like domain colouring the most! But Rienmann sphere is the coolest imo!

@harshpatel4431 2 года назад

Domain colouring is vey nice method.... Thanks

@LuanRHCP1 2 года назад

By all aleatory recomendations from RU-vid this was the best one so far. I'm a graduated enginner and this was so clarify. I do prefer 3D and vectors representation.

@mathemaniac 2 года назад

Thanks so much for the appreciation!

@StrunDoNhor 2 года назад

I totally thought this was a 3Blue1Brown video when I saw it in my recommended. Still very glad to have found you, nonetheless!

@little_bit_curious5122 2 года назад

Thank you for such nice explanation -)

@mathemaniac 2 года назад

Thanks for your compliment!

@devkar233 2 года назад

Nice video🙂

@carlosgaspar8447 2 года назад

Thanks!

@readjordan2257 Год назад

my favorite generally tends to be vector fields. i find it easiest to imagine trajectories and flow, it also tickles the art part of my mind and the rube goldberg/flash games/pinball/engineering/etc. intersection of my mind.

@griof 2 года назад

I recommend you another approach to visualization called X-ray. Essentially you plot two families of curves: im(z) = 0 and re(z)=0. Of course, when two lines of these families cross, it is a zero. They aren't super colorful visualizations, but they are very helpful. I recommend the paper "x-ray of Reimann zeta function. Arias-de-reyna, 2003" (cannot link to arxiv because of youtube censorship)

@mathemaniac 2 года назад

Technically, this can be done by the 3D Re-Im plot: Re(z) = 0 is where the surface touches the plane z = 0, and Im(z) = 0 is where the colour is red (or the colour that represents 0 in the plot); but if we are very specific to finding zeros, then maybe this can be useful.

@Happy_Abe 2 года назад

There is probably nothing like this visually on RU-vid Amazing video, great job!

@mathemaniac 2 года назад

Thanks for the appreciation!

@oscarfoley511 2 года назад

If you like these animations, i reccomend the og, 3b1b

@Happy_Abe 2 года назад

@@oscarfoley511 don’t have to recommend a channel I’ve been watching for around 4 years to me😄

@oscarfoley511 2 года назад

@@Happy_Abe then how can you say this visually on RU-vid??! :)

@Happy_Abe 2 года назад

@@oscarfoley511 was in regards to complex functions 3b1b hasn’t covered all these methods on his channel

@solapowsj25 2 года назад

Thanks for the detailed workout🏋🚴💪. ➖lines in a matrix are like lumens (light) of energy force. The lines at the North are exactly equal to the South in image kinetics. The frequency of oscillation in unit time in a specific area (2-D x-plane) is what determines whether the lines bound a Graviton? Gamma photon? UV, visible light, IR, or other rays.

@hamidkh5488 2 года назад

thank you . very interesting

@mathemaniac 2 года назад

Glad you enjoy the video!

@becbelk 2 года назад

great effort

@mathemaniac 2 года назад

Thanks a lot!

@MusicEngineeer Год назад

I think, the best way to visualize a complex function depends on the context. For example, in signal processing, when the complex function happens to represent an s- or z-domain transfer function of some system/filter, the 3D plot with modulus for the height is very useful. If the function is supposed to represent some geometric transformation, a z-w planes mapping visualization seems to be most useful. If I want to get an intuition for what a complex contour integral actually means, vector fields are the most appropriate visualization.

@code.with.chirag 2 года назад

Thank you!!!!!!!!!

@ramizhossain9082 2 года назад

Beautiful

@bhanupratapsingh6208 2 года назад

Thank you

@masatoedamura184 Год назад

Generally, I like to represent complex function as two z=f(x;y) surfaces one for real and one for im. basically 2 3d graphs. it's just way more natural than what you mentioned

@arthurmenezes5772 2 года назад

I remember messing with Geogebra last year and coming across i. I naturally started experimenting with it, and I always wondered why using it made such weird fractal shapes. Now I know! Thank you :) (btw the expression that generates the image in the timeline... I've seen it before. I'll answer here with the expression if I do find it. And yes, my pfp is x^i)

@mathemaniac 2 года назад

That function in the image in the timeline is a relatively simple one, and I am going to talk about it in the next video anyway.

@arthurmenezes5772 2 года назад

@@mathemaniac I forgot to see it, thanks :)