Exponential function

Exponential function or exp, can be defined as solution of differential equaiton

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 \exp^{\prime}(z)=\exp(z)}

with additional condition

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 \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 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 p} and 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 q} , the basic property holds:

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 \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 (syntax error): {\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:

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 \exp_b^0(z) =z }

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 \exp_b^1(z) =\exp_b(z) }

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 \exp_b^2(z) =\exp_b(\exp_b(z) }

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 \exp_b^{c+1}(z) =\exp_b(\exp_b^c(z) }

For non-integer values of 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 c} , the iterated exponential can be defined as

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 \exp_b^c(z) = \mathrm{sexp}_b\Big(c+ {\mathrm{sexp}_b}^{-1}(z)\Big) }

where 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 \mathrm{sexp}_b(z) } is function 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} satisfuing conditions

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+1)=\exp_b(F(z))}

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(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

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

and, within some range of values of 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 z}

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