The title is in quotation marks because it comes from the book Real and Complex Analysis by Walter Rudin. One should always be cautious around superlative statements of course, not just in mathematics but also in life, but in this case I think the wording is not without good reason. Rudin is referring to the function
which may be thought of as the constant
raised to the power of the argument . However, there is an even better definition, which is the one Rudin gives in the book:
written in more compact notation, this is
It can be shown that these two notions coincide. To emphasize its role as a function, is also often written as . It is also known as the exponential function.
We now explore some properties of . Let us start off with the case where is a real number. The function can be used to express any other function of the form
where is some nonzero real constant not necessarily equal to . By letting
where is the logarithm of to base (also known as the natural logarithm of ), we will then have
In other words, any function of the form where is some nonzero real constant can be expressed in the form simply by “rescaling” the argument by a constant.
For a real number, the function , which, as we have seen encompasses the other cases where is any nonzero real constant, is often used to express “growth” and “decay”. For instance, if we have a population which doubles every year, then after years we will have a population of , which, using what we have discussed earlier, can also be expressed as . If the population gets cut into half instead of doubling every year, we would then write it as .
But the truly amazing stuff happens when the argument of the exponential function is a complex number. Let us start with the case where it is purely imaginary. In this case we have the following very important equation known as Euler’s formula:
which for gives the result
The second equation, known as Euler’s identity, is often referred to by many as the most beautiful equation in mathematics, as it displays five of the most important mathematical constants in one equation: , , , , and .
For more general complex numbers with nonzero real and imaginary parts, we can use the rule for exponents
and treat them separately. The sine and cosine functions, aside from their original significance in trigonometry, are also used to represent oscillatory or periodic behavior. They are therefore useful in analyzing waves, which are a very important subject in both science and engineering. Equations such as the one above, combining growth and decay and oscillations, are used, for example, in designing shock absorbers in vehicles, which consist of a spring (which “fights back” the movement and oscillates) and a damper (which makes the movement “decay”).
There are certain technicalities regarding “multi-valuedness” that one must be wary of when dealing with complex arguments, but we will not discuss them for the time being (references are provided at the end of the post). Instead we will discuss a couple more properties of the exponential function.
First, we have the following expression for the exponential function as a product:
where is very big (which also means that is very small)
Using the language of limits in calculus, we can actually write
where is very big (which also means that is very small)
Historically, this is the motivation for the development of the exponential function, and the constant . Suppose that somewhere there is this greedy loan shark who loans someone an amount of one million dollars, at a rate of 100% interest per year. So our loan shark finds out that he can make more money by “compounding” the interest at the middle of the year, computing for 50% interest and adding it to the money owed to him, and then computing 50% of that amount again at the end of the year (reasoning that “technically” it is still 100% interest per year). So instead of one million dollars, he would be owed 1.5 million dollars by the second half of the year, and by the end of the year he would be owed 1.5 million dollars plus half of that, which is 0.75 million dollars, which makes for a total of 2.25 million dollars, much bigger than the 2 million he would have been owed without “compounding.”
So the greedy loan shark computes further and discovers that he can make even more money by compounding further; perhaps he can compound every quarter, computing for 25% interest after the first three months, adding it to the amount owed to him, after three months he again computes for 25% interest, and so on. He could make even compound a hundred times in the year, with 1% added every time he compounds the interest. So in his infinite greediness, let’s say our loan shark compounds an infinite number of times, in infinitely small amounts. What would be the amount owed to him at the end of the year?
It turns out, no matter how many times he compounds the interest, the money owed to him will never be greater than $2,718,281.83, or roughly “” million dollars, although it will approach that amount if he compounds it enough times. This quantity can be computed using the techniques of calculus, and in equation form it is exactly the expression for as a product that we have written above.
Finally, we give an important property of once again related to calculus, in particular to differential equations. We have discussed the notion of a derivative in An Intuitive Introduction to Calculus. An important property of the exponential function is that its derivative is equal to itself. In other words, if , then
This property plays a very important part in the study of differential equations. As we have seen in My Favorite Equation in Physics, differential equations permeate even the most basic aspects of science and engineering. The special property of the exponential function related to differential equations means that it appears in many laws of physics (as well as other “laws” unrelated to physics), and therefore its study is important to understanding these subjects as well.
Calculus by James Stewart
Real and Complex Analysis by Walter Rudin