Exponential function

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

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

with the additional condition

$\exp(0)=1$

The study of the exponential function began with Leonhard Euler around 1730 Since that time, it has had widely applications in technology and science; in particular, exponential growth is described with such functions.

Properties

The exponential is an entire function.

For any complex $p$ and $q$, the basic property holds:

$\exp(a)~\exp(b)=\exp(a+b)$

The definition allows to calculate all the derivatives at zero; so, the Taylor expansion has the 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 any complex $z$. In particular, the series converges for any real value of the argument.

Inverse function

The inverse function of the exponential is the 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 $\pi$:

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

When the logarithm has a cut along 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

The exponential is related to the trigonometric functions sine and cosine by de Moivre's formula:

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

Generalization of exponential

$\exp ^{c}(z)$ in the complex $z$ plane for some real values of $c$.

$\exp ^{c}(x)$ versus $x$ for some real values of $c$.

The notation $\exp _{b}$ is used for the exponential with scaled 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 $F$ satisfying conditions

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

$F(0)=1$

$F(z)~\mathrm {~is~holomorphic~and~bounded~at} ~|\Re (z)|<1$

The inverse function is defined with condition

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

and, within some range of values of $z$

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}

If in the notation 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} the superscript is omitted, it is assumed to be unity; for example 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=\exp_b} . If the suberscript is omitted, it is assumed to be 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{e}} , id est, 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^c=\exp_\mathrm{e}^c}

At 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 fixed points of logarithm 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 L\approx 0.31813150520476413 \pm 1.3372357014306895~ \mathrm{i}} are branch points of 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 \exp^c} ; in figure, the cut is placed parallel to the real axis. At 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<0} there is an additional cut which goes along the negative part of the real axis. In the figure, these cuts are marked with pink.

References

• Ahlfors, Lars V. (1953). Complex analysis. McGraw-Hill Book Company, Inc..
• H.Kneser. ``Reelle analytische Losungen der Gleichung 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 \varphi(\varphi(x))=\mathrm{e}^{x}} und verwandter Funktionalgleichungen. Journal fur die reine und angewandte Mathematik, 187 (1950), 56-67.