Exponential function

The exponential function of z, denoted by exp(z) or e^z, can be defined as the solution of the differential equation

$\exp ^{\prime }(z)\equiv {\frac {de^{z}}{dz}}=\exp(z)$

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 wide 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.

$\exp(z)=\lim _{n\rightarrow \infty }\left(1+{\frac {z}{n}}\right)^{n}~\forall z\in \mathbb {C}$

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}

Periodicity and relation with sin and cos functions

Exponential is periodic function; the period is $2\pi \mathrm {i}$:

$\exp(z+2\pi \mathrm {i} )=\exp(z)~\forall z\in \mathbb {C}$

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;

$\exp _{b}(z)=b^{z}=\exp \!{\Big (}\log(b)~z{\Big )}$

Notation $\exp _{b}^{c}$ is used for the iterated exponential:

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

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

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

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

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

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

$F(0)=1$

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

The inverse function is defined with condition

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

and, within some range of values of $z$

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

If in the notation $\exp _{b}^{c}$ the superscript is omitted, it is assumed to be unity; for example $\exp _{b}^{1}=\exp _{b}$. If the subscript is omitted, it is assumed to be $\mathrm {e}$, id est, $\exp ^{c}=\exp _{\mathrm {e} }^{c}$

At non-integer values of $c$, the fixed points of logarithm $L\approx 0.31813150520476413\pm 1.3372357014306895~\mathrm {i}$ are branch points of function $\exp ^{c}$; in figure, the cut is placed parallel to the real axis. At $c<0$ 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 $\varphi (\varphi (x))=\mathrm {e} ^{x}$ und verwandter Funktionalgleichungen. Journal fur die reine und angewandte Mathematik, 187 (1950), 56-67.