Calculators and complex numbers (Part 19)

In this series of posts, I explore properties of complex numbers that explain some surprising answers to exponential and logarithmic problems using a calculator (see video at the bottom of this post). These posts form the basis for a sequence of lectures given to my future secondary teachers.

To begin, we recall that the trigonometric form of a complex number z = a+bi is

z = r(\cos \theta + i \sin \theta) = r e^{i \theta}

where r = |z| = \sqrt{a^2 + b^2} and \tan \theta = b/a, with \theta in the appropriate quadrant. As noted before, this is analogous to converting from rectangular coordinates to polar coordinates.

Over the past few posts, we developed the following theorem for computing e^z in the case that z is a complex number.

Theorem. If z = x + i y, where x and y are real numbers, then

e^z = e^x (\cos y + i \sin y)

Example. Find all complex numbers z so that e^z = 5.

Solution. If z = x + iy, then

e^x (\cos y + i \sin y) = 5 (\cos 0 + i \sin 0)

Matching parts, we see that e^x = 5 and that the angle y must be coterminal with 0 radians. In other words,

x = \ln 5 \qquad and \qquad y = 2\pi n for any integer n.

Therefore, there are infinitely many answers: z = \ln 5 + 2 \pi n i.

Notice that there’s nothing particularly special about the number 5. This could have been any nonzero number, including complex numbers, and there still would have been an infinite number of solutions. (This is completely analogous to solving a trigonometric equation like \sin \theta = 1, which similarly has an infinite number of solutions.) For example, the complex solutions of the equation

e^z = -2 - 2i

are z = \ln 2\sqrt{2} - \displaystyle \frac{3\pi}{4} + 2 \pi n i.

These observations lead to the following theorems, which I’ll state without proof.

Theorem. The range of the function f(z) = e^z is \mathbb{C} \setminus \{ 0 \}.

Theorem. e^z = e^w \Longleftrightarrow z = w + 2\pi n i.

Naturally, these conclusions are different than the normal case when z is assumed to be a real number.

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For completeness, here’s the movie that I use to engage my students when I begin this sequence of lectures.



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