Elementary functions

We are now in a position to introduce many interesting functions. In particular, we define and obtain properties of the exponential and trigonometric functions.

Recall, that the rational functions that could not be integrated in terms of rational functions were of the form 1/(x - b) or (ax + b)/((x - c)⁺d²). The former is a continuous function when x $\neq$ b and the latter is continuous for all x. Thus we study the two functions

A(x) = $$\int_{1}^{x}$$$${\frac{dt}{t}}$$ andB(x) = $$\int_{0}^{x}$$$${\frac{dt}{1+t^2}}$$

The function A(x) is called the logarithm denoted by log(x) and the function B(x) is the inverse of the tangent function denoted by tan^-1(x) or arctan(x).

Exercise 61   Show that the function A(x) is monotonic and goes to infinity as x goes to infinity.

In particular, it follows that A has an inverse. This function is called the exponential function and denoted by exp(x). We can define the number e as exp(1); it is called Napier's natural base for the logarithm.

Exercise 62   Show that e is between 2 and 4.

Now the fundamental property of the logarithm is

Exercise 63   Show that A(x + y) = A(x) + A(y). Consequently we obtain exp(xy) = exp(x)exp(y).

This property and the monotonicity of exp characterise it.

Exercise 64   Let f (x) be a monotonic function so that f (xy) = f (x)f (y) and f (x) = a. Show that f (x) = exp(x log(a)).

Finally, we can define the hyperbolic functions as usual, cosh(x) = 1/2(exp(x) + exp(- x)) and so on.

We noe study the functions related with the function B.

Exercise 65   Show that the function B(x) is monotonic and remains bounded as x goes to infinity.

The least upper bound of the numbers 2B(x) is denoted by $\pi$. We can thus define the function tan(x) on the range (- $\pi$/2,$\pi$/2) as the inverse of B.

Exercise 66   Show that an arclength parametrisation of the unit circle x² + y² = 1 is given by t $\mapsto$ (sin(t), cos(t)) where we define these functions by the formulae

$${\frac{2\tan(x/2)}{1+\tan(x/2)^2}}$$ sin(x) =

$${\frac{1-\tan(x/2)^2}{1+\tan(x/2)^2}}$$ cos(x) =

Note that it follows that the perimeter of the circle x² + y² = 1 is 2$\pi$.

It is thus natural to extend these functions periodically for all t. We can define a multiplication on the unit circle by

(x₁, y₁)·(x₂, y₂) = (x₁x₂ - y₁y₂, x₁y₂ + x₂y₁)

Exercise 67   Show that the arclength parametrisation gives a group homomorphism from the additive group of real numbers to the circle.

This exercise provides the link between the trigonometric and exponential function which is expressed by the formula

exp($$\sqrt{-1}$$x) = cos(x) + $$\sqrt{-1}$$sin(x)

which the reader can try to prove!