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. Since that time, it has had wide applications in technology and science; in particular, exponential growth is described with such functions.
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.
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.
is widely used in applications; this notation is commonly accepted. Its approximate value is
Exponential is periodic function; the period is :
Generalization of exponential
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 logarithm on base e, notation ln is also used. In particular, , and so on.
- 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.