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 complex 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 $z\neq 0$, 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\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:

$\exp _{b}^{0}(z)=z$

$\exp _{b}^{1}(z)=\exp _{b}(z)$

$\exp _{b}^{2}(z)=\exp _{b}(\exp _{b}(z)$

$\exp _{b}^{c+1}(z)=\exp _{b}(\exp _{b}^{c}(z)$

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

$F(z+1)=\exp _{b}(F(z))$

$F(0)=1$

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(z)~ \mathrm{ ~is~ holomorphic~ and~ bounded~ in~ the~ range}~ |\Re(z)}<1}

The inverse function is defined with condition

$F{\Big (}F^{-1}(z){\Big )}=z$

and, within some range of values of $z$

$F^{-1}{\Big (}F(z){\Big )}=z$