Math 152-02 Lab 4/1/09 Allen Flavell and Michael Pelsmajer Introduction Most advanced mathematics software is well aware of the complex number system. In fact, it is likely that you have entered commands thinking only about real numbers, when the computer was treating them as complex numbers all along! In this lab, you will learn how to use Maple to take advantage of its ability to manipulate complex numbers. Representing Complex Numbers You can make a complex number very easily: just add i. Maple uses a capital I to represent i.Lowercase i does not work. point1 := 5+3*sqrt(-1); point2 := 8+2*I; Everything you can do with real numbers in Maple, you can do with complex numbers. point1+point2; limit(point2/x,x=point1); Two commands we will be using a lot in this lab are Re and Im. These commands take the real and imaginary components of a complex number, respectively. Re(point1); Im(point1); We can also represent complex numbers in polar coordinates. The Maple commands abs and argument do this: abs(point1); argument(point1); 1.a. Find the polar coordinate "r" for point2 without using "abs" or "argument". 1.b. Find it again, using "abs" or "argument" (pick the right one). 1.c. Find "theta" for point2. Declaring a complex function is also easy: just do it in the same way that you would declare a real function. If you pass a complex number into the function, Maple will use it correctly. For example: f := z -> (Re(z)+Im(z))/2; f(2); f(point1); Visualizing Complex Functions The plot command assumes that we are operating with a real function. plot(f(x),x=-5..5); So, in order to plot complex numbers, we will need to use commands from the plots package. restart:with(plots): When we deal with a real function, we are able to represent the function's behavior in its entirety using a two-dimensional graph. Consider the function g(x)=x2. f := x -> x^2; plot(f(x),x=-5..5); However, we cannot do this with complex functions. A complex function's domain or range cannot be represented using only a single axis (number line); an entire plane must be used. This means that to fully represent a complex function's behavior, we would have to use a four-dimensional space: two for the domain, and two for the range. One way of doing this would be to use a 3-dimensional plot and let the fourth dimension be time: L := solve(Im(f(x+y*I))=t,y) assuming x::real, y::real; x1 := x; x2 := L; animate(plot3d,[[x1,x2,Re(f(x1+x2*I))],x=-5..5,y=-5..5,axes=Boxed,grid=[10,10],labels=['Re(D)','Im(D)','Re(R)']],t=-5..5); In the above plot, we see the real and imaginary components of the domain (x1 and x2) as the horizontal axes, and we see the real component of range as height. As for the imaginary component of the range, we only see the points where the imaginary part t equals -5 (i.e., where 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 ). Now, click on the plot. You can rotate any plot by holding the mouse button and dragging. Try it. Now, click on the yellow triangle to see the animation. As time moves in the animation, we are taking different slices of the range, with the imaginary component equal to some number t, which goes from -5 to 5. (The function z2 is well-behaved enough that we will always get a curve for the function's values in these slices. A less-well behaved function like |z| would have very different results.) There's no special reason to make time correspond to the imaginary part of the range: Here's a plot where the domain is restricted to the real axis (the points LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIitGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5LUY2Ni1RJyZzZG90O0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTkZXLUYsNiVRIklGJ0YvRjJGOQ== ). Then as time t increases, the domain shifts (to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIitGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJ0RidGL0YyLUY2Ni1RJyZzZG90O0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTkZWLUYsNiVRIklGJ0YvRjJGOQ== ), by increasing the imaginary part of the domain. The restricted domain corresponds to one of the horizontal directions. The imaginary part of the range is shown as the other horizontal axis, and the real part of the range is shown as height (as before). animate(plot3d,[[x,Im(f(x+t*I)),Re(f(x+t*I))],x=-5..5,y=-5..5,axes=Boxed,grid=[10,10],labels=['Re(D)','Im(R)','Re(R)']],t=-5..5); (No questions about this part.) Let's clear the memory before continuing. Maple will forget about f, so we define it again. restart:with(plots): f := x -> x^2; Another option is to use a 3 dimensional plot and let the fourth dimension be color. The Maple command complexplot3d allows you to do this automatically. complexplot3d(f(z),z=-5-5*I..5+5*I, axes=boxed); In the above plot, we have the horizontal axes as the domain again, but we have the range displayed in polar: the magnitude is represented by height and the angle is represented as color. 2.a. How does the height compare when x and y are near zero (i.e., when we're near the middle of the horizontal plane) to when x and y increase or decrease (i.e., when we go away from that region)? 2.b. Calculate the modulus of 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, and simplify. (Do it by hand, or with Maple. Maple won't work unless you tell it that a and b are real, for which you need "assuming x::real, y::real".) 2.c. In the plot, modulus is supposedly shown as height. Explain why parts 2.a and 2.b agree with this. However, this sort of plot can be confusing, so another way would be to break the function into components first. In the next plot, we will use height for the real component and color for the imaginary component: complexplot3d([Re(f(x+y*I)), Im(f(x+y*I))],x=-5..5,y=-5..5, axes=boxed); While these methods do represent the function's behavior in its entirety, it is not practical to use them in the same way we use plots for Real functions. (Imagine trying to sketch one of those!) A different way of approaching the problem is to plot two planes, one corresponding to the domain and the other to the range, and "mark" certain points in the domain. The plot of the range would then display the "marked" points. Start with a few complex numbers: points := [[0,1],[2,1],[4,1]]; (This is a list of three points, and each point represents a single complex number, where the x-coordinate is the real part and the y-coordinate is the imaginary part.) Then we plug the points into f , which gives us new complex numberss. rangepoints := [[Re(f(points[1][1]+points[1][2]*I)),Im(f(points[1][1]+points[1][2]*I))], [Re(f(points[2][1]+points[2][2]*I)),Im(f(points[2][1]+points[2][2]*I))], [Re(f(points[3][1]+points[3][2]*I)),Im(f(points[3][1]+points[3][2]*I))]]; We plot the original points, then we plot f of the points. pointplot(points,title='Domain',labels=['Real','Imaginary'],color=["red","green","blue"],symbolsize = 20); pointplot(rangepoints,title='Domain',labels=['Real','Imaginary'],color=["red","green","blue"],symbolsize = 20); This process can be automated using the Maple procedure conformal. conformal takes a predetermined grid in the domain (determined by the grids option) and displays what this grid maps to in the range: conformal(x,x=-1+0*I..5+3*I,grid=[20,20], title='Domain', scaling=constrained); conformal(f(x),x=-1+0*I..5+3*I,grid=[20,20], title='Range', scaling=constrained); 3.a. What are the complex numbers at the four corners of the first grid? 3.b. Repeat the plot, but start with a grid shifted right by 2 units. 3.c. Repeat the plot, but start with a grid shifted up by 2 units. We can also perform mappings using different kinds of grids. Let's see what happens when we start with (part of) a polar grid: conformal(x,x=0-0*I..5+(Pi/3)*I,grid=[20,20], title='Domain', coords=polar, numxy=[100,100], view=[-5..5,-5..5]); conformal(f(x),x=-5-5*I..5+5*I,grid=[20,20], title='Range', coords=polar, numxy=[100,100], view=[-5..5,-5..5]); Great! Sometimes this gives us an intuitive picture of what the function is doing, in much the same way that we use plots of real functions. It could be even better: 3.d. Modify the "view" range in the second plot so that we can see the whole graph. How far away does it get from the origin, and why does this make sense for this particular complex function f ? (Clear memory again:) restart:with(plots): More Exercises 4. Using the information we've gathered plotting the function z2 so many different ways, what do you think the function is doing to the complex plane? 5.a. In class you were told that the exponential function also works with complex numbers. Plot ez using conformal. (Plot two grids: the start grid, and the function of the grid.) You may need to play with the grid and number of points in order to get a clear picture of what is going on. 5.b. Using the plot from part 5.a, what do you think the exponential function is doing to the complex plane? 5.c. When given only real numbers, the exponential function has the positive real numbers as its range. How is this reflected in the plot from part 5.a? 6.a. It is also possible to extend trigonometric functions in much the same way. Plot sin(z) and cos(z) using conformal. 6.b. How are these plots related to each other? Are they related to the plot of the exponential function? 6.c. Plot ez, sin(z), and cos(z) using complexplot3d. Use an axes option (such as axes=framed or axes=boxed) that will allow you to see which coordinate is which. 6.d. What relationships can you find between the functions? Do these relationships agree with what you found in part b? You may need to rotate your plots by clicking and dragging in order for some features to become obvious. Final Note Unlike most of the Maple Labs, this one was more about exploring complex functions, to get some quick looks at some advanced material. The course that really covers complex functions thoroughly (which would give you the background to better understand some of the things we've seen today) is Math 402: Complex Analysis.