We have seen that satisfies .

We will often leave the constant $h$ out of the notation and use terminology like for the “derivative”. It makes sense not to simplify to $x^n$ since the algebra structure is different.

Define the **exponential function** as

. It solves the equation . Because each of the approximating polynomials is monotone and positive also is monotone and positive for all . The fixed point equation reads so that for we have

where $e_n \to e$. Because $n \to e_n$ is monotone, we see that the exponential function depends in a monotone manner on h and that for the graphs of $\exp(x)$ converge to the graph of as .

Since the just defined exponential function is monotone, it can be inverted on the positive real axes. Its inverse is called . We can also define trigonometric functions by separating real and imaginary part of . Since , these functions satisfy and and are so both solutions to .