What does a complex function look like? #SoME3

have you ever wondered what a complex function looks like even though real functions have graphs that are easy to visualize it doesn't seem to be the case for a complex one there are actually many ways we can visualize these functions each of them brings something different to the table and in this video I'll explore then to see if any of them deserves to be called the graph of a complex function first things first what is the problem with trying to visualize a complex function the same way we visualize a real one when we make the graph of a real function we use two axes one for the domain the recovery in red this is where the values of the variable X are and we use another for the codomain that we're coloring in blue this is where the values y of X of the function are since we only need one dimension to represent a real number each of the axis takes one dimension so the graph as a whole takes up two dimensions which is perfect to draw on a surface however we need two Dimensions to represent a complex number one for the real part and another for the imaginary part so each axis would take up two dimensions and the graph as a whole needs four dimensions to be visualized unfortunately we live in a world with only three dimensions so this straightforward method is not possible let's look at an alternative here we have on the left the domain axis and on the right there's the codomain axis since these deal with complex numbers they need to be two-dimensional what a function does is take points from the domain and Maps them to points in the codomain let's see this in action with the function f of Z equals z to the fifth power plus z cubed we are using F to take points from the circle to the left into the curve to the right if we keep the circle fixed but change the function the curve to the right will also change here we have Z to the fourth power plus Z and z cubed minus Z the idea of the first visualization method is to consider a grid in the domain and looking at what happens after the grid gets mapped here we have Z squared the result allows us to understand how a square is deformed as we move it in the domain but this gets easier to interpret if we add some visual aids keeping that square is our example let's first get a closer look to make things less confusing a color in red the lines that are horizontal on the starting grid and our color in blue the vertical ones so starting from the origin Z equals zero we get mapped to the point F of zero taking a step in the red Direction Z increases by one and we get F of 1. taking a step in the blue Direction Z increases by I and we get F of I we could do this with a different starting point if Z equals 1 plus I then taking steps along the red curve gives us F of I and F of 2 plus I we've said increasing in One Direction and decreasing in the other similarly stepping along the blue curve gives us F of 1 and F of one plus two I this idea of stepping along curves is how you can read these graphs but of course this is not without its problems let's step back a little bit and now let's walk a little around the graph you can see when we got to -2 we got to the same point as when we passed through Z equals two this is not surprising because we're looking at the function Z square and the square of 2 equals the square of -2 it might not look like a big deal but this means that if we try to look at something a little bit more complicated than a simple power then things will get very complicated let's take another step back and see what I'm talking about we're still looking at that square now let's change it to z cubed minus Z if that looks confusing that is nothing check out Z to the fifth power plus Z squared it's clear that if we want to tackle more complicated functions we will need a simpler visualization method than mapping grids a different approach called domain coloring tries to solve the problem of fitting a four-dimensional graph into something understandable by looking for Dimensions elsewhere instead of spatial Dimensions we will use color dimensions every pixel in your screen has a color defined by a precise mixture of red green and blue RGB these quantities can be thought as three coordinates meaning that we can arrange all of these colors in a cube borrowing some Dimensions from this color space we reduce the amount of spatial Dimensions did it conveyed information in the graph but the color Dimensions used for the main coloring are normally not red green and blue what's usually used instead is the Hue of the collar and how bright it is to understand how this is done we must recall that all complex numbers can be written in a polar form where instead of the real and imaginary Parts the number is described by its absolute value and its angle fat what the main coloring does is encode the angle as the Hue and the absolute value as the brightness visually it looks something like this here the arrow is pointing to a complex number and if we extend it to the disk we get the color representing that value so as the arrow spins the angle of the complex number changes and together with it the Hue representing it if on the other hand the absolute value changes the color keeps its Hue but changes brightness normally smaller absolute values are represented by darker colors and larger absolute values are represented by Brighter colors so we know the function is close to zero if the caller is close to Black and we know the function is big if the caller is close to White a downside is that because of the way colors work the brightness is either too low or too high the Hue becomes harder to tell a different way of using brightness to denote absolute value is adding contour lines if you ever use the map that had indications of elevation those lines that tell you how high or low a place is are contour lines the idea is to do the same but for absolute values instead of height now that's enough talk let's see how the main coloring Works in practice we'll start with the function Z divided by Z square plus 1. that's a pretty picture but what is this as the name implies we're looking at the domain of the function in particular the real and imaginary axes are going through here so the points in the screen have the coordinates of Z what about the callers these represent the value of the function different Hues represent different angles so Reds and blues have different angles same for yellows and greens in particular if we follow the real line we see that a pure red corresponds to a positive real number while cyan is a negative real number and this yellowish green is a number with no real part and a positive imaginary part similarly purple is a number with no real part and a negative imaginary part we can also see the root of the function in Black at Z equals zero as well as the two points where the function blows up in white at Z equals I and Z equals minus I as I said another way of exploring the brightness is with counter lines in particular these contour lines and using have a bit of bleeding the blurrier the line the slower the absolute value is changing we can still see the root of the function and the point where we have blow ups because the Contours Bunch up around those points let us move on and take a look at Z Square here we can see explicitly that the square of Z equals the square of minus Z because if we rotate the figure by 180 degrees around the origin we'd get the same exact image another interesting example is the exponential function the angle here repeats periodically if we move up this is because the angle of exponential of Z is just the imaginary part of Z so moving along the imaginary axis is like moving around a circle walking along the real axis on the other hand we see equally spaced counter lines this is because the absolute value of exponential of Z is the exponential of the real part of Z and I'm using a log scale for the counters this however shows a weakness of drawing unlabeled Contours like these if we were to go back to encoding the absolute value as brightness we'd get a completely different picture that better shows how exponential of that grows but you also lose a lot of the Precision you get with counters one possible solution is to combine both ideas let us leave exponential of Z and move to a different function this is z to the fifth power plus Z Square we can see here its roots and the fact that it diverges when the absolute value becomes very large the way it diverts gets clearer if we add contour lines we can now see that it becomes very homogeneous as we get farther from the origin moving on to something more complicated this is the tangent of Z here the counters show us that the absolute value changes mostly close to the real axis and the brightness that's a Stell apart what are roots and what are blow ups the graph also shows us that the tangent of Z converges when the imaginary part of Z goes to infinity or to minus infinity namely it converges to I and minus I something similar happens when we look at Z minus for I divided by Z Plus for I if it's root at 4i and a blow up at minus for I let us take a step back and think about what we're doing for a bit I've shown it two different ways that we can make a graph of a complex function and in a sense they are opposite to each other when we mapped grids we were registering Zed at F upset that is the information is which values are carried to a given point by a function f the main coloring on the other hand registers F upset at Zed information is the value that F takes at a given point now we have seen these methods are not perfect let me highlight some problems and see what we can improve the main issue with mapping grids is that the lines crossing each other get very confusing this is inevitable byproduct of registering Zed instead of f of Z on the other hand even using only two color Dimensions the main color impacts way too much information in colors one possible Improvement is that since we can spare a third spatial Dimension then we only really need to use one color dimension if we do this right we can keep the general Spirit of the main coloring but making the colors easier to interpret let's see what happens if we use the Z coordinate for the absolute value and keep using the Hue for the angle let's start with something very simple F of Z equals z if we were to add access to this it would look like this but I'll leave the note so that we can just move around and explore the shapes we can see as expected the absolute value growing linearly as we move away from the origin and spinning the surface we see the different Hues representing the different angles if we look from the top we'll see something close to the usual domain coloring let's move on to Z squared we can see now that adding a third dimension allows us to have a much more precise idea of the absolute value than we ever could with domain coloring even using Contours let's take a look at Z to the fifth power plus Z Square it's clear now that the function is much steeper than Z Square which also wasn't obvious with just domain coloring finally let's look at an example where there is a blow up Z minus I divided by Z plus I we can clearly see the blow up at Z equals minus I as well as the root at Z equals I it shows up like a kink in the surface now I want to move to a final question any of these visualizations the graph of a function first what is the technical definition of a graph well the graph of a real function is actually defined by this a set of points X and f of x were X spanning the whole real line but how do we translate this into something more tangible suppose we make the plot of a function for every horizontal coordinate X we have a corresponding point with vertical coordinate f of x so the point in the curve is X and f of x but this means that the graph of the function is the curve but fought off as a set for a complex function the graph is defined in a similar way but we must take into account that both the variable and the function have a real and an imaginary part so the graph will be the points X Y representing the parts of the complex variable Z and U of X and Y B of X and Y representing the parts of the function value F of Z so for a real function the graph is an object in two Dimensions having one parameter X the result is a curve in the plane a complex function on the other hand has a graph that is four dimensional but has two parameters X and Y and this results in a surface in some four-dimensional space so if you want to truly see the graph of a complex function we need a way to see a surface in four dimensions this is actually a very common problem and the main two for that is called projection this is how we are able to see things like decline bottle or the boy surface projection is actually very simple it can be thought of as rotating and translating an object and then just getting rid of some of the coordinates actually the grid mappings we started with are projections of the graph where we got rid of the first two coordinates let us instead get rid of the last one so we'll use the Z coordinate for the real part of the function this is z squared we can now see an upwards Parabola if we move along the Rio line and the downwards one if we move along the imaginary one this is z cubed and this is z minus I divided by Z plus I comparing this to what we had using the absolute value as the vertical coordinates we see a completely different way that the blow up looks like we can now see that we can reach the blow up in two different ways with a positive real part or with a negative real part depending on the side that we choose this fact is only registered in the colors of the previous plots so it's very easy to miss it finally we can add back the fourth dimension that we dropped by coloring the projection you could either color it according to the imaginary part of the function or according to the angle of the function which is what I'm doing so that we have an easier time comparing with the previous plots let's try for example the tangent of Z if I recalling and comparing it with the domain coloring that we had earlier and let's now just finish with Z squared I hope you found this interesting and thanks for watching

Channel: mathematimpa

Views: 119,930

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Id: r1h3eNQ2YM0

Length: 20min 38sec (1238 seconds)

Published: Fri Aug 18 2023