Revision as of 04:37, 29 October 2008 by imported>Dmitrii Kouznetsov
Exponential function or exp, can be defined as solution of differential equation
![{\displaystyle \exp ^{\prime }(z)=\exp(z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/61da01ebe27f04128076482661f1b750421009f9)
with the additional condition
![{\displaystyle \exp(0)=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8cef12440a3f192bc8a9a29ae266c225dac6151c)
The study of the exponential function began with Leonhard Euler around 1730[1]
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
and
, the basic property holds:
![{\displaystyle \exp(a)~\exp(b)=\exp(a+b)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02d3dad6df902f970e5e988ede7a2863e97b50f4)
The definition allows to calculate all the derivatives at zero; so, the Taylor expansion has the form
![{\displaystyle \exp(z)=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}~~\forall z\in \mathbb {C} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f1eeb99b9d9663dbc15bacf10a5900bd6fa80e9)
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.
![{\displaystyle \exp(z)=\lim _{n\rightarrow \infty }\left(1+{\frac {z}{n}}\right)^{n}~\forall z\in \mathbb {C} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e381070de8891da28fb3c4f044ebff0895084ded)
Inverse function
The inverse function of the exponential is the logarithm; for any complex
, the relation holds:
![{\displaystyle \exp(\log(z))=z~\forall z\in \mathbb {C} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8696f10a11c93ad49ba2645b2bb459fbf08708e0)
Exponential also can be considered as inverse of logarithm, while the imaginary part of the argument is smaller than
:
![{\displaystyle \log(\exp(z))=z~\forall z\in \mathbb {C} ~\mathrm {~such~that~} |\Im (z)|<\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d39c3b6d020d88109e26491f37c496484e0ec143)
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
- 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
:
![{\displaystyle \exp(z+2\pi \mathrm {i} )=\exp(z)~\forall z\in \mathbb {C} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/96b262ead6adda58d37d59eaede5268caab89f4e)
The exponential is related to the trigonometric functions sine and cosine by de Moivre's formula:
![{\displaystyle \exp(\mathrm {i} z)=\cos(z)+\mathrm {i} \sin(z)~\forall z\in \mathbb {C} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/52909a1c896492896e42477ba98d27d7eb5130af)
Generalization of exponential
![{\displaystyle \exp ^{c}(z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03a291b153a9e34566058d56b75333fc7126e6be)
in the complex
![{\displaystyle z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98)
plane for some real values of
![{\displaystyle c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455)
.
![{\displaystyle \exp ^{c}(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a09905310021786698896916a2bf4a77d5ace628)
versus
![{\displaystyle x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
for some real values of
![{\displaystyle c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455)
.
The notation
is used for the exponential with scaled argument;
![{\displaystyle \exp _{b}(z)=b^{z}=\exp \!{\Big (}\log(b)~z{\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3976892c93fbc4bf9aa46bb6928d41a19ff139b9)
Notation
is used for the iterated exponential:
![{\displaystyle \exp _{b}^{0}(z)=z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29c2dd9930bab5a29cb47a8fc717cec6f59d2c4a)
![{\displaystyle \exp _{b}^{1}(z)=\exp _{b}(z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebdbbf2c7d486afc0e15546f7121b6605d5bf49e)
![{\displaystyle \exp _{b}^{2}(z)=\exp _{b}\!{\Big (}\exp _{b}(z){\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e8f8203f7cc86fbe3a1037b3bf3a6fb0c4f6b22)
![{\displaystyle \exp _{b}^{c+1}(z)=\exp _{b}\!{\Big (}\exp _{b}^{c}(z){\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/092e205454888634c717b0e08a7a3adead063ca3)
For non-integer values of
, the iterated exponential can be defined as
![{\displaystyle \exp _{b}^{c}(z)=\mathrm {sexp} _{b}\!{\Big (}c+{\mathrm {sexp} _{b}}^{-1}(z){\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fb2509e96ae2f44ebf534c60d17308e82bb32ec)
where
is function
satisfying conditions
![{\displaystyle F(z+1)=\exp _{b}\!{\Big (}F(z){\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be4e27c8f772c0df27bbad53f6e77c147ad78604)
![{\displaystyle F(0)=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fe64b0aee8c3456f57db4924b822de6538b4489)
![{\displaystyle F(z)~\mathrm {~is~holomorphic~and~bounded~at} ~|\Re (z)|<1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/58bb9b240efcda36f29b4cb61f136877e710883d)
The inverse function is defined with condition
![{\displaystyle F{\Big (}F^{-1}(z){\Big )}=z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90ba48467ad50b4c848dd44c2d891967a4fe11cc)
and, within some range of values of
![{\displaystyle F^{-1}{\Big (}F(z){\Big )}=z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1154ded761447f9ce1ca9975d402f34638675bde)
If in the notation
the superscript is omitted, it is assumed to be unity; for example
. If the suberscript is omitted, it is assumed to be
, id est,
At non-integer values of
, the fixed points of logarithm
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, these cuts are marked with pink.
References
- ↑ 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.