Computational lab on complex numbers

Transcription

1 Computational lab on complex numbers SAK, physics 1140 March 15, Objective To learn how to use MATLAB as an advanced calculator and gain familiarity with complex numbers using MATLAB. 2 Introduction Complex numbers have many uses in physics and engineering. Examples we will explore here are how the are plotted in the real-imaginary plane. This is based upon the hand-out given on complex numbers. Generally calculations with complex numbers are painful and tedious. However with MATLAB it is no more difficult than doing calculations with real numbers. Here you will learn how to use MATLAB as a calculator and for making plots. ( In other words an advanced scientific calculator. ) This is just a fraction of the power of MATLAB, because you can do so much more like circuit design, simulations, data acquisition etc. This assignment can be submitted by typing the answers into the script you develop below (use the % to make a comment that will not interfere with the code). Then the completed script to us. 3 Procedure You will be working in pairs. One member is primarily responsible for the typing and is designated base. The other member helps the base, and if stuck seeks help from other groups or TAs. This person is designated the 1

2 runner, and can roam whereas the base cannot. The two will electronically submit a lab report, which is the code, also called m-file. Start MATLAB and locate the command window. Here try typing in simple calculations, such as pi+exp(4). Upon hitting return MAT- LAB will evaluate that line. Also try defining variables, by typing m=3, a = 2, F= m*a In fact, when numerically evaluating formulas I often declare the variables (e.g. q = 1.602E-19, g=9.81 etc.) and then let MATLAB do the plugging and chugging. Now open the editor, create a new blank m-file, which is where you will type the script below: function complex_numbers1 c = 1+sqrt(-1) %this is a complex number, note that sqrt(-1) = i real(c) %this is the real part of c imag(c) %this is the imaginary part of c c = 1+sqrt(-100) %this is a complex number, note that sqrt(-1) = i real(c) %this is the real part of c imag(c) %this is the imaginary part of c To evaluate this script, press F5, or select run. As you see, the script evaluates just as if you had entered this in the command line. It s just easier here to edit and rerun. Next you will make a plot by appending these lines of code: figure(1) plot( c, ro ) %the symbol for plotting is a red circle title( plotting a complex number in the plane ); %% adding to the plot c2 = i %preserving original points when additional points are plotted plot( c2, ks ) %the symbol used is black square %% adding complex numbers is like vector addition plot( c+c2, pb ) %symbol is a blue pentagon 2

3 Again hit F5 to run this longer script. Figure out how complex numbers are plotted in the plane. You might want to adjust the axes using the command axis( [ ] ). Check yourself by trying different values for c2, i.e. c2 = pi+sqrt(2)*1i. So far so good. Next you will see how curves are plotted. Type figure(2) t = linspace( 0, 2*pi ) plot( t, cos(t), -r ); plot( t, sin(t), -b ); xlabel( t ); title( trigonometric functions ); QUESTION 1: What sort of trig function is plotted? You provide the answer by completing this in the script: %%%%%%%%%%%%%%%answer question 1 %The trig. function is a %%%%%%%%%%%%%%%% Now let s proceed to the third figure. Also, find out how to insert breakpoints, and step through the program. Use either the on-line help or send out your runner. figure(3) t = linspace( 0, 10 )%assigning a 1X100 vector to t c = -1 plot( t, exp(c*t), r- ) %red line title( plots involving the exponential function ) c = 1E-1 plot( t, exp(c*t), k-- ) %dashed black line c = 5.0i plot( t, exp(c*t), b-. ) %blue dot-dash line c = i plot( t, exp(c*t), k: ) %green dotted line xlabel( t ); 3

4 Pay attention to the warnings when you have run the script. Replace the command with the commented blue line with plot( t, real( exp(c*t)), b- ) %blue dot-dash line and also try plot( t, imag( exp(c*t)), b- ) %blue dot-dash line QUESTION 2: What happens when MATLAB is asked to plot complex numbers? Note the answer depends on whether one complex numbers is supplied, as for the black squares, or two number arguments are supplied, as for the blue dot-dashed line. Again the answer is provided by completing this in the script: %%%%%%%%%%%%%%%answer question 2 %Complex numbers are %%%%%%%%%%%%%%%% The fourth figure is getting interesting, because here we will plot a function in the complex plane. Type in figure(4) t = linspace( 0, 10 )%assigning a 1X100 vector to t c = 1i plot( exp(c*t), r- ) %red line title( plots involving the exponential function ) axis image %setting the aspect ratio to 1 c = -.1+1i plot( exp(c*t), k-- ) %red line c =.1+1i plot( exp(c*t), b-. ) %red line axis image %setting the aspect ratio to 1 Make sure you understand how the circle, inwards and outwards spirals are formed, because you ll need it for the last assignment. 4

5 QUESTION 3: Create a spiral that starts at 1+0i, and spirals inwards to.1+0i, making 4 turns. Hints: First draw a single circle - what are the limits of the t-vector to do this? Now can you draw a circle with four turns? If so, you are almost done... figure(5) t = linspace( 0,?) c =? plot( exp(c*t) ) 5