Exponential function

Exponential function or exp, can be defined as solution of differential equaiton
 * $$ \exp^{\prime}(z)=\exp(z)$$

with additional condition
 * $$ \exp(0)=1 $$

Exponential function is believed to be invented by Leonarf Euler some centuries ago. Since that time, it is widely used in technology and science; in particular, the exponential growth is described with such function.

Properties
exp is entire function.

For any comples $$p$$ and $$q$$, the basic property holds:
 * $$ \exp(a)~\exp(b)=\exp(a+b) $$

The definition allows to calculate all the derrivatives at zero; so, the Tailor expansion has form
 * $$ \exp(z)=\sum_{n=0}^\infty \frac{z^n}{n!} ~ ~ \forall z\in \mathbb{C} $$

where $$\mathbb{C}$$ means the set of complez numbers. The series converges for and complex $$z$$. In particular, the series converge for any real value of the argument.

Inverse function
Inverse function of the exponential is logarithm; for any complex $$z\ne 0$$, the relation holds:


 * $$ \exp(\log(z))=z ~ \forall z\in \mathbb{C} $$

Exponential also can be considered as inverse of logarithm, while the imaginary part of the argument is smaller than $$\pi$$:


 * $$ \log(\exp(z))=z ~ \forall z\in \mathbb{C} ~ \mathrm{~ such ~ that ~ } |\Im(z)|<\pi $$

While lofarithm has cut at the negative part of the real axis, exp can be considered

Number e
$$\mathrm {e} = \exp(1)$$ is widely used in applications; this notation is commonly accepted. Its approximate value is


 * $${\rm e}=\exp(1) \approx 2.71828 18284 59045 23536$$

Relation with sin and cos functions

 * $$ \exp(\mathrm{i} z) = \cos(z)+\mathrm{i} \sin(z) ~ \forall z\in \mathbb{C} $$

Generalization of exponential
Notation $$\exp_b$$ is used for the exponential with modified argument;


 * $$\exp_b(z)=b^z=\exp(\log(b) z)$$

Notation $$\exp_b^c$$ is used for the iterated exponential:
 * $$ \exp_b^0(z) =z $$
 * $$ \exp_b^1(z) =\exp_b(z) $$
 * $$ \exp_b^0(z) =\exp_b(\exp_b(z) $$
 * $$ \exp_b^{c+1}(z) =\exp_b(\exp_b^c(z) $$

For non-integer values of $$c$$, the iterated exponential can be defined as
 * $$ \exp_b^c(z) =

\mathrm{sexp}_b\Big(c+ {\mathrm{sexp}_b}^{-1}(z)\Big) $$ where $$ \mathrm{sexp}_b(z) $$ is function $$F$$ satisfuing conditions


 * $$F(z+1)=\exp_b(F(z))$$
 * $$F(0)=1$$
 * $$F(z)~ \mathrm{ ~is~ holomorphic~ and~ bounded~ in~ the~ range}~ \Im(z)<1$$

The inverse function is defined with condition
 * $$F\Big(F^{-1}(z)\Big)=z$$

and, within some range of values of $$z$$
 * $$F^{-1}\Big (F(z)\Big)=z$$