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, $\exp ^{c}=\exp _{\mathrm {e} }^{c}$

Function $f=\exp ^{c}(z)$ is shown in figure with levels of constant real part and levels of constant imaginary part. Levels $\Re (f)=-3,-2,-1,0,1,2,3,4,5,6,7,8,9$ and $\Im (f)=-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14$ are drown with thick lines. Red corresponds to a negative value of the real of imaginaryt part, black corresponds to zero and blue corresponds to the positeive values. Levels $\Re (f)=-0.2,-0.4,-0.6,-0.8$ are shown with thin red lines. Levels $\Im (f)=0.2,0.4,0.6,0.8$ are shown with thin green lines. Levels $\Re (f)=\Re (L)$ and Levels $\Im (f)=\Im (L)$ are marked with thick green lines, where $L\approx 0.31813150520476413+1.3372357014306895~\mathrm {i}$ is fixed point of logarithm. At non-integer values of $c$, the fixed points of logarithm $L$ and $L^{*}$ 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, the cuts are marked with pink lines.

For real values of the argument, function $y=\exp ^{c}(x)$ is ploted in figure versus $x$ for values $c=0,\pm 0.1,\pm 0.5,\pm 0.9,\pm 1,\pm 2$.
In particular, $\exp ^{-1}(x)=\ln(x)$ and $\exp ^{-2}(x)=\ln {\big (}\ln(x){\big )}$.

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.