# Exponential function

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The exponential function of , denoted by  or , can be defined as the solution of the differential equation



with the additional condition



The study of the exponential function began with Leonhard Euler around 1730.[1] 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:



The definition allows to calculate all the derivatives at zero; so, the Taylor expansion has the form



where  means the set of complex numbers. The series converges for any complex . 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 , the relation holds:



Exponential also can be considered as inverse of logarithm, while the imaginary part of the argument is smaller than :



When the logarithm has a cut along the negative part of the real axis, exp can be considered.

## Number e

 is widely used in applications; this notation is commonly accepted. Its approximate value is



## Periodicity and relation with sin and cos functions

Exponential is periodic function; the period is :



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



## Generalization of exponential

(CC) Image: Dmitrii Kouznetsov
 in the complex  plane for some real values of .
(CC) Image: Dmitrii Kouznetsov
 versus  for some real values of .

The notation  is used for the exponential with scaled argument;



Notation  is used for the iterated exponential:






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



where  is function  satisfying conditions





The inverse function is defined with condition



and, within some range of values of 



If in the notation  the superscript is omitted, it is assumed to be unity; for example . If the subscript is omitted, it is assumed to be , id est, 

Function  is shown in figure with levels of constant real part and levels of constant imaginary part. Levels  and  are drown with thick lines. Red corresponds to a negative value of the real or the imaginaryt part, black corresponds to zero, and blue corresponds to the positeive values. Levels  are shown with thin red lines. Levels  are shown with thin green lines. Levels  and Levels  are marked with thick green lines, where  is fixed point of logarithm. At non-integer values of ,  and  are branch points of function ; in figure, the cut is placed parallel to the real axis. At  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  is ploted in figure versus  for values .
in programming languages, inverse function of exp is called log.

For logarithm on base e, notation ln is also used. In particular, ,  and so on.

## References

1. William Dunham, Euler, the Master of us all, MAA (1999) ISBN 0-8835-328-0. Pp. 17-37.
• Ahlfors, Lars V. (1953). Complex analysis. McGraw-Hill Book Company, Inc..
• H.Kneser. Reelle analytische Losungen der Gleichung  und verwandter Funktionalgleichungen. Journal fur die reine und angewandte Mathematik, 187 (1950), 56-67.