Math 221

Homework #4

Instructions. This is due on October 30 at the beginning of class. It is worth 20 points. Late papers lose 3 points per day or partial day late. You must work by yourself, but feel free to consult with me. Explain what you did and why you did it: good explanations are essential! Your paper should start with an introduction and finish with a conclusion. Start early, ask questions and do well.

Introduction. In the 1600’s when calculus was being invented, Isaac Newton and the gang often used infinite series to represent functions (see the historical note on Brook Taylor on page 514). Ever since, series have provided one of the basic means of understanding complicated ideas. In this exercise, we use series to make sense out of several expressions involving complex numbers. Recall that i represents the square root of -1. Then it follows that i² = -1, i³ = -i, i⁴ = 1, i⁵ = i and so on. You will use these values below.

Problems. Start by writing out the Taylor series (c = 0) for e^x, sinx and cosx.

The question is what could be meant by e^ix. Of course, an imaginary power has no obvious meaning. We will derive its meaning from Taylor series. Simply substitute in ix for x in the Taylor series for e^x. Simplify this expression using the powers of i listed above. Notice that half the terms contain i and half don’t. Separate the terms that do from the ones that don’t. Then identify one group of terms as the Taylor series for sinx and the other group of terms as the Taylor series for cosx. Write out your final result for e^ix. This is known as Euler’s formula.

Use Euler’s formula to evaluate e^ip/4. When engineers at MIT shout: We’re number e^2pi ! what do they mean? Show that e^pi + 1 = 0 and note that this equation consists of the five most famous numbers in mathematics.

Recall that the cosine and sine are periodic functions with period 2p. It turns out that in complex numbers the exponential function is also periodic. Use Euler’s formula to show that for f(x) = e^x, you have f(x+2pi) = f(x). Use this equation to identify the period of the exponential function.