suppose that we can substitute an arbitrary complex number z = x + iy in the exp

Question

suppose that we can substitute an arbitrary complex number z = x + iy in the expression for ex, and that the usual rules for exponentiation apply. Then

ex+iy =exeiy =ex(cosy+isiny).

In particular, if r is a positive real number, then elnr+i =r(cos+isin).

Thus every nonzero complex number z = r(cos + i sin ) has a logarithm. In fact the possible solutions to ew = z are

w = ln r + ( + 2n)i, n an integer.

We call any of these values a logarithm of z and write w = log z. Of course, log is not a well-defined function. Note that, for real x, the exponential function ex is one-to-one and its values are the positive real numbers; hence ln x is defined for positive x. For complex z, the exponential function is no longerone-to-one:ez1 =ez2 exactlywhenz2=z2+2niforsomeintegern, and the values of the complex exponential are all nonzero complex numbers. Thus log z is “defined” for all z = 0, but only up to adding an arbitrary integer multiple of 2i.

For example, log(1) = log(ei) = i + 2ni = (2n + 1)i, for any integer n.

Explanation / Answer

When the Matlab plot function sees a complex vector as its first argument, it understands the components to be points in the complex plane. So the octagon in the left half of figure 8.3 can be defined and plotted using e i with theta = (1:2:17)’*pi/8 z = exp(i*theta) p = plot(z); The quantity p is the handle to the plot. This allows us to complete the graphic with set(p,’linewidth’,4,’color’,’red’) axis square axis off An exercise asks you to modify this code to produce the five-pointed star in the right half of the figure. Once we have defined e i for real , it is clear how to define e z for a general complex z = x + iy, e z = e x+iy = e x e iy = e x (cos y + isin y) Finally, setting z = i, we get a famous relationship involving three of the most important quantities in mathematics, e, i, and e i = 1 Let’s check that Matlab and the Symbolic Toolbox get this right. >> exp(i*pi) ans = -1.0000 + 0.0000i 12 Chapter 8. Exponential Function >> exp(i*sym(pi)) ans = -1 Recap %% Exponential Chapter Recap % This is an executable program that illustrates the statements % introduced in the Exponential Chapter of "Experiments in MATLAB". % You can access it with % % exponential_recap % edit exponential_recap % publish exponential_recap % % Related EXM programs % % expgui % wiggle %% Plot a^t and its approximate derivative a = 2; t = 0:.01:2; h = .00001; y = 2.^t; ydot = (2.^(t+h) - 2.^t)/h; plot(t,[y; ydot]) %% Compute e format long format compact h = 1; while h > 2*eps h = h/2; e = (1 + h)^(1/h); disp([h e]) end %% Experimental version of exp(t) t = rand s = 1; term = 1; n = 0; r = 0; while r ~= s 13 r = s; n = n + 1; term = (t/n)*term; s = s + term; end exp_of_t = s %% Value of e e = expex(1) %% Compound interest fprintf(’ t yearly monthly continuous ’) format bank r = 0.05; y0 = 1000; for t = 0:20 y1 = (1+r)^t*y0; y2 = (1+r/12)^(12*t)*y0; y3 = exp(r*t)*y0; disp([t y1 y2 y3]) end %% Payments for a car loan y0 = 20000 r = .10 h = 1/12 n = 36 p = (1+r*h)^n/((1+r*h)^n-1)*r*h*y0 %% Complex exponential theta = (1:2:17)’*pi/8 z = exp(i*theta) p = plot(z); set(p,’linewidth’,4,’color’,’red’) axis square off %% Famous relation between e, i and pi exp(i*pi) %% Use the Symbolic Toolbox exp(i*sym(pi))

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