# Hyperbolic functions.

Definition 4.3.1

 . . .

Example 4.3.2

Proposition 4.3.3

1. .
2. .

Example 4.3.4 (A nice equation)   We solve the equation .

Let , where . Then:

1. We have now:

2. We take care of the second equation:

3. We substitute into the first equation:
1. . This equation has no real solution.
2. . This equation is equivalent to

Its (real) solutions are and .
3. . This equation has no solution.

In conclusion, the complex solutions of the equation are:

 and

Example 4.3.5 (A nice equation - second way)   We solve the equation .

We substitute and solve for the equation . The solutions are: and . In order to find the corresponding values of , we need logarithms. They will be defined in the next paragraph (v.i. 4). This exercise will be finished in example  4.6.

Noah Dana-Picard 2007-12-24