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.
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.
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.
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.
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 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.
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!
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
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.”
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
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!
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.
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.
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.
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!
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....
@Mr Fl0v 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.
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.
@@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.
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!
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.
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
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.
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.
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).
@@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 ?
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
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.
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.
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.
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?
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 ;)
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
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.
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!!!
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 :)
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.
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.
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.
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.
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)
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.
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.
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.
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.
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.
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.
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 :)
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.
Im hugely biased towards the w-z plane. They are most satisfying and easy to understand. Its almost gives the same visualisation as the the x-y graph. Having two planes is rather more joyful. (The only problem it has is that it diverts you from the fact that complex functions are actually functions of the whole plane, and not specific lines/shapes. We aren't able to get what the function does to the 'plane', and hence dont get to know the real essence of the function.)
In 3:34 the answer is that the root at top left (2+i) is double, has a multiplicity of 2, which you can see because has 2 full cycles of colour and also in the formula, this root has an exponent of 2
By all aleatory recomendations from TH-cam this was the best one so far. I'm a graduated enginner and this was so clarify. I do prefer 3D and vectors representation.
As an engineer, I prefer the Vector Fields but for " filtering " work the 3D plots could be useful. When dealing with solutions for Maxwell's equation it is mentally satisfying to see the Electric and the Magnetic field " curling each other in waveguides and cavity resonators operating at different modes. With fluid flows as in boats and propellers and aircraft, I prefer the Vector fields. When I go out for a walk in the countryside, and I see a field of wheat with blowing wind, I see those wheat stalks swaying with the wind. While walking alone in Scotland at the Cheviot Hills I often saw small tornadoes and the dust and debris show the vectors swirling around, At about 1963 I used to go to Pitlochry to see the salmon going up the water pipes to bypass the dams which were too high for them. TO help salmon have a rest, along the pipe they made higher volume rooms so that the flow will slow down " at the corners" and the salmon used to know and feel these vectors and knew where to rest till it got its " energy back" I used to use the ferry from Glasgow to the Isle of Aran and I stayed on deck feeding the seagulls while they glided and soared on the wind uplift on the side of the Ferry and holding their stationary position feeding from my hand. With my wife, before we got married we used to go to the waterfalls dropping off the ridges of a valley, and with the wind blowing in the opposite direction the water drops floated in and around the dropping edge, With the white sunlight entering the floating water droplets, we would see a rainbow and all this was within a few meters around us. Again I am an engineer, and I love to see the effect of the vector field around me. I also took my wife sailing in hard blowing winds and the tufts I had on the sails showed and told all the story of the pressure and velocity field around the sail. In my country, many children find difficulty in mathematics because many teachers teach mathematics in class and many teachers do not even know that mathematics is what models the universe around us. Unfortunately, many teachers still write the numbers as the alphabet with " rounded curves" ),1,2,3,4,5,6,7,8,9, unlike the original Arabic form where the number symbol contains the " corners " which tell is value as "2" should be written as " Z" note two angles less than 90 degrees, while "3" should be written as (M of W oriented by 90 degrees) as the symbol contains three corner angles, 8 should not be rounded as shown but 8 should be written as two squared one above the other. Another situation where I prefer to use a vector field is to show how antenna radiate due to an acceleration field, BY putting a small pail full of water on top of a wooden broom stick, when the stick oscillates in a flat plane, the " water" is radiated when the acceleration field is high enough, When the pail is rotated on the vertical stick then angular acceleration will be generated and this represents the radiation from an antenna. With phased antennas or water vibrations or light slits, even diffraction and som many " vectors can be brought to life. Congratulations on these presentations, I did all this 70 years ago when we did not have digital computers and all I had was a 30 cc moped with which I roamed all over England and Scotland and my own country, seeking to see vector in their reality. On occasions, I drove from Dover cliffs to see the " winds" up to P{itlockrhry to see the salmon and in my country with my wife and daughter and now my grandson whenever it rains and it is windy, we go and see the gravity, wind, fields suspending the water droplets while the vibration in the electromagnetic light will produce the colored rainbow in the floating water droplets. Note that as an engineer I modified the R + jW plane from two to three dimensions where the third dimension is the " Phase " as in electrical engineer the complex frequency R+jW as in ( e^( R+jW)t can have a phase shift A which the conventional complex X-Y plane does not show. Again congratulations for showing us on a computer how beautiful is the world around us, and all this is within a 20-mile radius of my home village!!!! I prefer living with the real engineering vectors, but here in the comforts of my home, it is an honor and a privilege to see younger brilliant minds, depicting the computers as they should. Oh, how I wish that primary school teachers will start the numerals as they should be, with cornered symbols denoting their value. So many people treat the numbers symbols as if they are writing poetry and the children think that they need to " remember and memorize" the Poem rather than the " real acticvity and real logic and real reasoning and rational thinking between the RELATIONS THAT DEPICT THE TRUTH AND REALITY, all around us. Congratulations once again for these mathematical representations.
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)
the colored, animated pólya vector field, mod‐arg 3D plot, and the z-w plane transformation animation makes the most sense to me. I'm glad there's the mod-arg plot bc you can just take snapshots of different parts of it, therefore it could be printed in books without losing "the point" (the info you want to point out). Not everyone uses animations to learn, so it's cool that there's options (i prefer animation).
Maths + this font can't stop me from thinking of 3B1B. No disrespect though, your content is awesome. Maybe collab in the near future? And what's your accent? :D
In my 20s I developed two methods to solve multidimensional problems through visualization that are not any of these. These were practical problems. The first problem had to do with modeling, analyzing, measuring and synthesizing acoustic fields. I had no formal training in acoustics but I had the tools I needed and as I see it the solution was straightforward. A single source at an arbitrary point in an acoustic space radiates sound at different intensities and in different directions. The problem is solved by imagining a sphere or other closed surface Ia sphere is best because the normal vector through any area dS is unique) through which the sound passes. The amplitude can vary with time in any direction. For simplicity take all of the directions to radiate at the same frequency with different fixed amplitudes. Problem, how would you arrive at the field at another point in space? The problem is made more complex by the fact that multiple reflections will arrive from the same directions at different times. A problem arises because different reflections at different frequencies will arrive with different spectral changes with time. The solution to first normalize the first arriving sound so that its arrival time is at zero milliseconds and its amplitude is at a reference say zero db. Then the solution is to superimpose the field from each arriving vector component on a sphere where the arrival time is left in the time domain and the amplitude change and spectral change from the normalized first sound is plotted as a three dimensional plot. Do this for each frequency in the audible range and you have the source function at one point mapped into the arriving function at another point. Since this explains the relationship between acoustic energy propagated at one point and the consequential field arriving at another point I call the theory Acoustic Energy Field Transfer Theory and the mapping function Acoustic Energy Field Transfer Function. This understanding allowed me to design acoustic fields at will. A simplified variant of a machine to do this I call an Electronic Environmental Acoustic Simulator. You can see the concept in my patent 4,332,979. Look at figure 5 (ignore figures 1,2, 3a, and 3b. They relate to the measurement scheme the patent office said was a different invention. The figures were not deleted inadvertently.) It would have been better had I drawn the curves starting along the time axis and falling off with increasing frequency. I have what is probably the only working prototype in the world. The second problem had to do with how to objects collide. The more I thought about it the more impossible it seemed. I imagined a microscope that could zoom into the point of nearest approach and slow time down to a crawl or even freeze it. The objects never come into physical contact. In fact they are mostly empty space. The valence electrons can repel by their electrostatic force or they can bond and stick together but they never touch. Now as a senior in electrical engineering I memorized for my final exam every equation in the textbook Fano Chu Adler and Lai, nearly 250 of them and could solve all kinds of problems with them. All of Maxwell's equations (no shortcuts or lumped symbols) in the curl and divergence form as integral and as partial differential equations but they told me nothing about how or why they worked. What frustration, I realized I knew nothing about how or why, I only knew what would happen. So I started thinking about atoms and subatomic particles, matter energy, time space, stuff I was supposed to know in the first two years of physics. And then one day I got an idea. I imagined the particles as we still believe them, tiny billiard balls. I lined up the z axis pointing straight at me and then turned it 90 degrees so that the z axis disappeared and the w axis appeared. And lo and behold I was in a very different universe. They are in that dimension nothing like I had ever imagined. Perhaps one day I'll publish my theory after I pay up my dues to get back into the American Institute of Physics. That was nearly 50 years ago. Suddenly a lot of things became clear, a theory which explains a lot of things I didn't understand at all. Is the theory right? I don't have the slightest idea but it all comes from one assumption, that photons must have mass. How do I know that? Several ways but the most compelling one is that when matter turns into energy mass disappears and photons appear in their place. Where did they come from? The mass. That is why matter and energy can be converted into each other, matter is made up of photons. How? That's what the model explains.
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
Ha! I asked my tutor in analysis how a complex function could be mapped, he said it's just a function with an input and output. Okay. :/ Later on I visualized two 2D planes in 3D space, each one representing the complex plane and arrows directing to it's destination. So I like the vector space idea, it's pretty close to what I thought intuitively!
3:25 The black spot on the left corresponds to the zero of (z+1), the one to the right of it to (z-1) and the black spot with a double color wheel period corresponds to the squared term (z-2-i)^2. The 2 white spots come from division by the zeros of z^2+2+2i.
Beautiful ❤️
Thank you so much Mithuna!
Unfortunately they have many more distractions too.
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.
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.
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.
Awww shit! You gonna cover my boi Möbius!?
@@jacobhoward7579 Of course! But in a future video.
Check this out. Alot of people say trippy but it's so complex its not understandable my a none math person.
th-cam.com/video/uH5CFPUXT9U/w-d-xo.html
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.
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!
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.
@@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.
Is this the same (or similar) of f(x) = x^2 having two roots both being "0"?
@@Xingchen_Yan yes, it's a squared term so you'll see this in the plot as a double winding of the colors
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!
Thank you so much!
yeah, the aerofoil mapping example was really great
I studied engineering and with this video, complex analysis and complex algebra finally made sense to me.
Glad to help!
@@mathemaniac Thank you for this marvelous work ❤️
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.
3:17 The zero has multiplicity 2. The hue goes through two full cycles as you go around it.
Yep - that's true!
Genius!
Thanks. I suspected the function isn't injective around that point, because of the square, but couldn't put my finger on the word.
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
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.”
Yes - that's also an option!
"This looks like-" A PRINGLE!! "an airfoil." oh...
Wonderful video and explanation as always!
Haha, now that you pointed it out, it does look a little like a pringle :)
Exactly my reaction
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!
Glad you like the video!
My favorite complex plottings in descending order:
1. Domain colouring.
2. Mod-Arg plot.
3. Re-Im plot.
4. Streamplot / Pólya Vector Field.
To me, domain coloring seems like the most versatile option, but a z-w plane in the context of subsets is probably my favorite.
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
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!
I like the first (colour) mapping the best. It is the most natural, neutral, and easy to visualise :)
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.
Thanks so much for the appreciation!
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.
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.
@@NexusEight Yay ! Harolds of the world unite ! Thanks for the appreciative remarks.
This series of complex analysis is gold mine. You revolutionized the way of teaching high dimensional abstract maths.
Ways to visualize? More like “Facts explained beautifully before my eyes!” This is an incredible series and I can’t wait to watch more.
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!
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....
> _shows that there is only one point at infinty...._
nice. yeah, +inf and -inf are coinciding that is? 😅
@Mr Fl0v 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.
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.
You made me change my major. Probably best decision I have ever made.
Thanks
Wow! Haven't thought that I could have such an impact for such a small channel!
@@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.
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!
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.
@@yash1152 zoom in for detail?
Função de Complexo em Complexo. 5 representações gráficas! Espetacular!
I loved the vectors representation . Even though I am new and couldn't catch all , but glad to watch .
Nice work
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
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.
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.
Glad you like them!
Great video! I'm excited to learn a lot more about complex functions than the little I know already. Awesome animations and clear descriptions.
Thanks! Glad you liked it!
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).
I like vector fields the most because they're really pretty! :D but all of these are super helpful tools for visualizing complex functions
Yes - all of them are very useful tools of visualisation!
@@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 ?
We live in a 3D society
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
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.
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.
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.
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?
The Riemann Sphere corresponds nicely to cosmology. If an n-dimensional sphere is projected onto an m-dimensional cartesian manifold where m
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 ;)
Grande vídeo, grandes animações e grande conhecimento por trás delas.
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
I won't regret subscribing this channel. Amazing work, man.
Thanks for the appreciation and the subscription!
I liked your cap :))
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.
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!!!
Thank you for your immense hard work . I have started to love maths because of teachers like you
this video made my jaw drop, especially with the joukowsky transform, for me who's studying aircraft engineering it surely was fascinating
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 :)
P.S.: I love the z-w plane method the most, I think it is the most obvious for seeing the conformal mapping.
There will be more videos! And yes, thanks for appreciating my efforts in making this video!
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.
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.
Man you completely outdid yourself with this one keep it up
Thanks for the kind words!
My favorite would be the domain coloring. Beautiful
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.
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.
I have that book “Visual Complex Analysis”. My favorite book on the subject.
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)
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.
Vector fields do feel more homely. However each method is useful for some purposes.
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.
Love love love your content, I'm amazed at how prolific you are!
Glad you enjoy it!
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.
Love how the airfoil shape is seen to be a particularly angled view of the pringle outline shape, thanks to your animation 😮
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.
Wonderfully explained! My compliments. Also thank you for spelling colour correctly ❤️
Thanks for the compliment!
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.
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.
Subbed and Belled! Great content about something i’ve been recently very interested about, can’t wait to learn more about complex functions
Thanks for your appreciation!
I’m currently watching this in my college math class. Nice job!!!!❤
Great work! Came across your channel for the first time, loved it very much! Keep up the good work and keep the videos coming! :)
Thanks for your appreciation!
Great animations and detailed explanations! Extremely helpful! Thank you!
Glad it was helpful!
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.
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 :)
my favorite is the mod-arg 3d plot, it contains the most information and in a very understandable manner
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.
Domain coloring is clearly the most beautiful. Dont' know which one is most useful
Im hugely biased towards the w-z plane. They are most satisfying and easy to understand. Its almost gives the same visualisation as the the x-y graph. Having two planes is rather more joyful. (The only problem it has is that it diverts you from the fact that complex functions are actually functions of the whole plane, and not specific lines/shapes. We aren't able to get what the function does to the 'plane', and hence dont get to know the real essence of the function.)
There is probably nothing like this visually on TH-cam
Amazing video, great job!
Thanks for the appreciation!
If you like these animations, i reccomend the og, 3b1b
@@oscarfoley511 don’t have to recommend a channel I’ve been watching for around 4 years to me😄
@@Happy_Abe then how can you say this visually on TH-cam??! :)
@@oscarfoley511 was in regards to complex functions
3b1b hasn’t covered all these methods on his channel
In 3:34 the answer is that the root at top left (2+i) is double, has a multiplicity of 2, which you can see because has 2 full cycles of colour and also in the formula, this root has an exponent of 2
Domain colouring is vey nice method.... Thanks
By all aleatory recomendations from TH-cam this was the best one so far. I'm a graduated enginner and this was so clarify. I do prefer 3D and vectors representation.
Thanks so much for the appreciation!
As an engineer, I prefer the Vector Fields but for " filtering " work the 3D plots could be useful.
When dealing with solutions for Maxwell's equation it is mentally satisfying to see the Electric and the Magnetic field " curling each other in waveguides and cavity resonators operating at different modes.
With fluid flows as in boats and propellers and aircraft, I prefer the Vector fields. When I go out for a walk in the countryside, and I see a field of wheat with blowing wind, I see those wheat stalks swaying with the wind. While walking alone in Scotland at the Cheviot Hills I often saw small tornadoes and the dust and debris show the vectors swirling around, At about 1963 I used to go to Pitlochry to see the salmon going up the water pipes to bypass the dams which were too high for them. TO help salmon have a rest, along the pipe they made higher volume rooms so that the flow will slow down " at the corners" and the salmon used to know and feel these vectors and knew where to rest till it got its " energy back"
I used to use the ferry from Glasgow to the Isle of Aran and I stayed on deck feeding the seagulls while they glided and soared on the wind uplift on the side of the Ferry and holding their stationary position feeding from my hand. With my wife, before we got married we used to go to the waterfalls dropping off the ridges of a valley, and with the wind blowing in the opposite direction the water drops floated in and around the dropping edge, With the white sunlight entering the floating water droplets, we would see a rainbow and all this was within a few meters around us.
Again I am an engineer, and I love to see the effect of the vector field around me. I also took my wife sailing in hard blowing winds and the tufts I had on the sails showed and told all the story of the pressure and velocity field around the sail.
In my country, many children find difficulty in mathematics because many teachers teach mathematics in class and many teachers do not even know that mathematics is what models the universe around us.
Unfortunately, many teachers still write the numbers as the alphabet with " rounded curves" ),1,2,3,4,5,6,7,8,9, unlike the original Arabic form where the number symbol contains the " corners " which tell is value as "2" should be written as " Z" note two angles less than 90 degrees, while "3" should be written as (M of W oriented by 90 degrees) as the symbol contains three corner angles, 8 should not be rounded as shown but 8 should be written as two squared one above the other.
Another situation where I prefer to use a vector field is to show how antenna radiate due to an acceleration field, BY putting a small pail full of water on top of a wooden broom stick, when the stick oscillates in a flat plane, the " water" is radiated when the acceleration field is high enough, When the pail is rotated on the vertical stick then angular acceleration will be generated and this represents the radiation from an antenna. With phased antennas or water vibrations or light slits, even diffraction and
som many " vectors can be brought to life.
Congratulations on these presentations, I did all this 70 years ago when we did not have digital computers and all I had was a 30 cc moped with which I roamed all over England and Scotland and my own country, seeking to see vector in their reality. On occasions, I drove from Dover cliffs to see the " winds" up to P{itlockrhry to see the salmon and in my country with my wife and daughter and now my grandson whenever it rains and it is windy, we go and see the gravity, wind, fields suspending the water droplets while the vibration in the electromagnetic light will produce the colored rainbow in the floating water droplets.
Note that as an engineer I modified the R + jW plane from two to three dimensions where the third dimension is the " Phase " as in electrical engineer the complex frequency R+jW as in ( e^( R+jW)t can have a phase shift A which the conventional complex X-Y plane does not show.
Again congratulations for showing us on a computer how beautiful is the world around us, and all this is within a 20-mile radius of my home village!!!! I prefer living with the real engineering vectors, but here in the comforts of my home, it is an honor and a privilege to see younger brilliant minds, depicting the computers as they should.
Oh, how I wish that primary school teachers will start the numerals as they should be, with cornered symbols denoting their value. So many people treat the numbers symbols as if they are writing poetry and the children think that they need to " remember and memorize" the Poem rather than the " real acticvity and real logic and real reasoning and rational thinking between the RELATIONS THAT DEPICT THE TRUTH AND REALITY, all around us. Congratulations once again for these mathematical representations.
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)
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.
@@mathemaniac I forgot to see it, thanks :)
Nice video! I really appreciate the great animations you've created.
Glad you like them!
Thank you. Please continue the series.
Thanks for your appreciation!
Riemann spheres. The other perspectives provide interesting insights into concrete visualizations of complex functions.
I like domain colouring the most! But Rienmann sphere is the coolest imo!
All those animations on 3b1b suddenly started to make more sense. Different ways to visualise the info.
the colored, animated pólya vector field, mod‐arg 3D plot, and the z-w plane transformation animation makes the most sense to me.
I'm glad there's the mod-arg plot bc you can just take snapshots of different parts of it, therefore it could be printed in books without losing "the point" (the info you want to point out). Not everyone uses animations to learn, so it's cool that there's options (i prefer animation).
Maths + this font can't stop me from thinking of 3B1B. No disrespect though, your content is awesome. Maybe collab in the near future?
And what's your accent? :D
In my 20s I developed two methods to solve multidimensional problems through visualization that are not any of these. These were practical problems.
The first problem had to do with modeling, analyzing, measuring and synthesizing acoustic fields. I had no formal training in acoustics but I had the tools I needed and as I see it the solution was straightforward. A single source at an arbitrary point in an acoustic space radiates sound at different intensities and in different directions. The problem is solved by imagining a sphere or other closed surface Ia sphere is best because the normal vector through any area dS is unique) through which the sound passes. The amplitude can vary with time in any direction. For simplicity take all of the directions to radiate at the same frequency with different fixed amplitudes. Problem, how would you arrive at the field at another point in space? The problem is made more complex by the fact that multiple reflections will arrive from the same directions at different times. A problem arises because different reflections at different frequencies will arrive with different spectral changes with time. The solution to first normalize the first arriving sound so that its arrival time is at zero milliseconds and its amplitude is at a reference say zero db. Then the solution is to superimpose the field from each arriving vector component on a sphere where the arrival time is left in the time domain and the amplitude change and spectral change from the normalized first sound is plotted as a three dimensional plot. Do this for each frequency in the audible range and you have the source function at one point mapped into the arriving function at another point. Since this explains the relationship between acoustic energy propagated at one point and the consequential field arriving at another point I call the theory Acoustic Energy Field Transfer Theory and the mapping function Acoustic Energy Field Transfer Function. This understanding allowed me to design acoustic fields at will. A simplified variant of a machine to do this I call an Electronic Environmental Acoustic Simulator. You can see the concept in my patent 4,332,979. Look at figure 5 (ignore figures 1,2, 3a, and 3b. They relate to the measurement scheme the patent office said was a different invention. The figures were not deleted inadvertently.) It would have been better had I drawn the curves starting along the time axis and falling off with increasing frequency. I have what is probably the only working prototype in the world.
The second problem had to do with how to objects collide. The more I thought about it the more impossible it seemed. I imagined a microscope that could zoom into the point of nearest approach and slow time down to a crawl or even freeze it. The objects never come into physical contact. In fact they are mostly empty space. The valence electrons can repel by their electrostatic force or they can bond and stick together but they never touch. Now as a senior in electrical engineering I memorized for my final exam every equation in the textbook Fano Chu Adler and Lai, nearly 250 of them and could solve all kinds of problems with them. All of Maxwell's equations (no shortcuts or lumped symbols) in the curl and divergence form as integral and as partial differential equations but they told me nothing about how or why they worked. What frustration, I realized I knew nothing about how or why, I only knew what would happen. So I started thinking about atoms and subatomic particles, matter energy, time space, stuff I was supposed to know in the first two years of physics. And then one day I got an idea. I imagined the particles as we still believe them, tiny billiard balls. I lined up the z axis pointing straight at me and then turned it 90 degrees so that the z axis disappeared and the w axis appeared. And lo and behold I was in a very different universe. They are in that dimension nothing like I had ever imagined. Perhaps one day I'll publish my theory after I pay up my dues to get back into the American Institute of Physics. That was nearly 50 years ago. Suddenly a lot of things became clear, a theory which explains a lot of things I didn't understand at all. Is the theory right? I don't have the slightest idea but it all comes from one assumption, that photons must have mass. How do I know that? Several ways but the most compelling one is that when matter turns into energy mass disappears and photons appear in their place. Where did they come from? The mass. That is why matter and energy can be converted into each other, matter is made up of photons. How? That's what the model explains.
I totally thought this was a 3Blue1Brown video when I saw it in my recommended. Still very glad to have found you, nonetheless!
Awesome video! Thank you!
3:33 the upper right zero is a double root since it has 2 copies of all of the colors around it, this is equivalent to the (z-2-i)^2
Yes exactly!
It’s the multiplicity of the zero. That black spot cycles through two color wheels
Yes!
Can and gonna binge watch the full series. A request‐ please do a video about tensors.
Great video really enjoyed...An artistic and aesthetic demonstration
Thanks for the appreciation!
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
Domain colouring, followed by vector fields for specific applications.
I salute peoples who do these kind of maths
3D plot is definitely my favorite
Amazing! Congratulations !! This is such a massive amount of work
That’s great😊 Never know some ways before, learn it now❤
Oooh, the reimann sphere is excellent
Enjoy all but the z->w animation is fav.
z-w planes being my favorite, or it's at least what cones to my mind when I think of what complex numbers are & do.
Ha! I asked my tutor in analysis how a complex function could be mapped, he said it's just a function with an input and output. Okay. :/ Later on I visualized two 2D planes in 3D space, each one representing the complex plane and arrows directing to it's destination. So I like the vector space idea, it's pretty close to what I thought intuitively!
3:25 The black spot on the left corresponds to the zero of (z+1), the one to the right of it to (z-1) and the black spot with a double color wheel period corresponds to the squared term (z-2-i)^2. The 2 white spots come from division by the zeros of z^2+2+2i.
Yes, exactly!