Chapter 5: Elementary Functions
Definition 5.1.1: If a is a positive real number, then the exponential function with base a is the function .
We have seen that the exponential function with base a is defined for all rational numbers x. Furthermore, we know that:
Proposition 5.1.1: Let a > 0 and x and y be rational numbers. Then
So far, we have not defined for any x other than the rational numbers. Let x be an arbitrary real number. Since the exponential function is an increasing function, if c/d < x < e/f where c/d and e/f are rational numbers, then . By taking pairs of numbers rational numbers closer and closer to x, one can get values of and which are as close to each other as one pleases. So, it would be natural to conjecture that
Conjecture: For any real number x, there is precisely one real number y such that for all rational numbers c/d and e/f with c/d < x < e/f.
The conjecture is true, but hard to prove. In this course, we will assume that it is true and define to be the uniquely defined y specified in the conjecture. So, the domain of the exponential function is now the set of all real numbers. Furthermore, we will assume without proof that
Proposition 5.1.2: The assertions of Proposition 5.1 hold true for all real numbers x. Furthermore, the range of the exponential function is the set of all positive real numbers.
A function f has been defined as a rule which assigns to each element x of its domain a uniquely defined value f(x) in its range. The function is defined to be one-to-one or injective if two different values of the domain are never assigned the same value in the range, i.e. f(a) = f(b) only if a = b. If the function is one-to-one, it follows that for every y in the range, there is a uniquely defined x in the domain such that f(x) = y. This means that we have a function g defined by g(y) = x provided that f(x) = y.
Definition 5.2.1 If f is a one-to-one function with domain D and range R, then the inverse of f is the function g with domain R and range D defined by g(y) = x if and only if f(x) = y.
Example 5.2.1: Here are some examples:
The graph of a function f is the set of all the points (x, f(x)) for x in the domain of f. If g is the inverse of f, the g(y) = x if and only if f(x) = y. So, the graph of g is the set of all points (y, g(y)) = (f(x), x) for y in the domain of g (or x in the domain of x). You can interpret this as saying that the graph of the inverse g of f is obtained by reflecting the graph of the function f across the 45 degree line y = x. If you have the graph on a sheet of paper, you can see the graph of the inverse function by turning the paper over and rotating it 90 degrees -- try it with your favorite one-to-one function.
Definition 5.3: Let be a real number with and . The logarithm with base is the inverse function of the exponential function with base .
Proposition 5.3.1: Let a, x, and y be positive real numbers and t be any real number. The logarithm function has the following properties:
Proof: Assertion (i) is true simply because the logarithm function is the inverse of the exponential function.
(ii) Suppose that x < y and for some positive x and y. Since the exponential function is increasing, we know that . But, since the two functions are inverses, this just says that contrary to hypothesis.
(iii) One has because of the addition formula for the exponential function. Further and because the logarithm function is the inverse of the exponential function. So, . But this means that since the logarithm function is the inverse of the exponential function. The proofs of the remaining assertions follow similarly using the corresponding properties of the exponential function.
Another way of defining an exponential function is to define
The terms are of the form for k = 0, 1, .... and the sum infinite. The idea is that the terms become arbitrarily small in such a manner that the sum of the first n terms for large n approaches a well defined real number. Of course, this has not been proved; again, we will simply assume that this is true, leaving the proof for a future course. In fact, we will assume that this defines an exponential function with base The approximate value of e is 2.71828182845904523536028747135266249775724709369995957496697. It can be shown that e is not a rational number, in fact, it like the number does not satisfy any polynomial equation with rational number coefficients.
The function is so important in applications that it is referred to as the exponential function; whenever you hear of an exponential function without reference to a base, this is the one it refers to. The inverse of the exponential function is called the natural logarithm and is denoted .
Because the factorials in the denominators of the terms of the series that defines grow so rapidly, the sums of the first n terms very rapidly approach the value of the exponential function. So, the series is useful in creating tables of values of the exponential function.
Until now, we have been dealing with the exponential function as a function of a real variable. One can extend the domain to the set of all complex numbers. Of course, there are lots of ways of doing this, but following the philosophy of this course, we would like to do it in a manner that preserves all the important properties of the exponential function. We cannot hope to preserve the increasing property -- since the complex numbers are not an ordered field. However, we certainly would like to preserve the all important addition formula. We have two different approaches:
Again, without proof, both definitions can be shown to be equivalent and to yield the same function. Furthermore, the all important addition formula is preserved. In fact, this unifies the trigonometry with the study of the exponential function -- we see that the addition formulas of the exponential function and of trigonometry are just special cases of the addition formula for the complex exponential function.
Proposition 5.4.1: For any complex numbers and , one has .
Proof: Let and . Using the second definition, one has . Now substituting the addition formulas for the real exponential function, the sine function and the cosine function, and expanding out the results, one sees that this is identical to the result of expanding out .
Exercise 5.4.1: Use the first definition and the binomial theorem to give another proof of the addition formula.
Exercise 5.4.2: Use and the series definition of the exponential function to show that
Exercise 5.4.3: Use the series from the last exercise to approximate values of the sine and cosine function for various values of x. For example, how many terms does it take to approximate to five decimal places?
Exercise 5.4.4: Prove Euler's formula: .
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