Exponential function: Difference between revisions
imported>Dmitrii Kouznetsov (New page: '''Exponential function''' or exp, can be defined as solution of differential equaiton : <math> \exp^{\prime}(z)=\exp(z)</math> with additional condition : <math> \exp(0)=1 </math> Expon...) |
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''' | {{subpages}} | ||
: <math> \exp^{\prime}(z)=\exp(z)</math> | The '''exponential function''' of <math>z</math>, denoted by <math> \exp(z)</math> or <font style="vertical-align:+10%;"><math>e^z</math></font>, can be defined as the solution of the differential equation | ||
with additional condition | : <math> \exp^{\prime}(z)\equiv \frac{d e^z}{dz}=\exp(z)</math> | ||
: <math> \exp(0)=1 </math> | with the additional condition | ||
: <math> \exp(0)=1.\, </math> | |||
The study of the exponential function began with [[Leonhard Euler]] around 1730.<ref>William Dunham, ''Euler, the Master of us all'', MAA (1999) ISBN 0-8835-328-0. Pp. 17-37.</ref> | |||
Since that time, it | Since that time, it has had wide applications in technology and science; in particular, [[exponential growth]] is described with such functions. | ||
is described with such | |||
==Properties== | ==Properties== | ||
The exponential is an [[entire function]]. | |||
For any | For any complex ''p'' and ''q'', the basic property holds: | ||
: <math> \exp(a)~\exp(b)=\exp(a+b) </math> | : <math> \exp(a)~\exp(b)=\exp(a+b) </math> | ||
The definition allows to calculate all the | The definition allows to calculate all the derivatives at zero; so, the [[Taylor expansion]] has the form | ||
: <math> \exp(z)=\sum_{n=0}^\infty \frac{z^n}{n!} ~ ~ \forall z\in \mathbb{C} </math> | : <math> \exp(z)=\sum_{n=0}^\infty \frac{z^n}{n!} ~ ~ \forall z\in \mathbb{C} </math> | ||
where <math>\mathbb{C}</math> means the set of | where <math>\mathbb{C}</math> means the set of [[complex number]]s. | ||
The series converges for | The series converges for any complex <math>z</math>. In particular, the series converges for any real value of the argument. | ||
: <math>\exp(z)=\lim_{n\rightarrow \infty}\left(1+\frac{z}{n}\right)^n ~ \forall z\in \mathbb{C}</math> | |||
==Inverse function== | ==Inverse function== | ||
The inverse function of the exponential is the [[logarithm]]; for any complex <math>z\ne 0</math>, the relation holds: | |||
: <math> \exp(\log(z))=z ~ \forall z\in \mathbb{C} </math> | : <math> \exp(\log(z))=z ~ \forall z\in \mathbb{C} </math> | ||
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: <math> \log(\exp(z))=z ~ \forall z\in \mathbb{C} ~ \mathrm{~ such ~ that ~ } |\Im(z)|<\pi </math> | : <math> \log(\exp(z))=z ~ \forall z\in \mathbb{C} ~ \mathrm{~ such ~ that ~ } |\Im(z)|<\pi </math> | ||
When the logarithm has a cut along the negative part of the real axis, exp can be considered. | |||
==Number e == | ==Number e == | ||
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: <math>{\rm e}=\exp(1) \approx 2.71828 18284 59045 23536</math> | : <math>{\rm e}=\exp(1) \approx 2.71828 18284 59045 23536</math> | ||
== | ==Periodicity and relation with [[sin]] and [[cos]] functions== | ||
Exponential is [[periodic function]]; the period is <math>2 \pi \mathrm i </math>: | |||
: <math> \exp(z+2\pi \mathrm{i})=\exp(z) ~ \forall z\in \mathbb{C} </math> | |||
The exponential is related to the [[trigonometric function]]s [[sine]] and [[cosine]] by ''[[de Moivre]]'s formula'': | |||
: <math> \exp(\mathrm{i} z) = \cos(z)+\mathrm{i} \sin(z) ~ \forall z\in \mathbb{C} </math> | : <math> \exp(\mathrm{i} z) = \cos(z)+\mathrm{i} \sin(z) ~ \forall z\in \mathbb{C} </math> | ||
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==Generalization of exponential== | ==Generalization of exponential== | ||
{{Image|Sqrt(exp)(z).jpg|right|500px|<math>\exp^c(z)</math> in the complex <math>z</math> plane for some real values of <math>c</math>.}} | |||
{{Image|Expc.jpg|right|200px|<math>\exp^c(x)</math> versus <math>x</math> for some real values of <math>c</math>.}} | |||
The notation <math>\exp_b</math> is used for the exponential with scaled argument; | |||
: <math>\exp_b(z)=b^z=\exp(\log(b) z)</math> | : <math>\exp_b(z)=b^z=\exp\!\Big(\log(b)~ z\Big)</math> | ||
Notation <math>\exp_b^c</math> is used for the iterated exponential: | Notation <math>\exp_b^c</math> is used for the iterated exponential: | ||
: <math> \exp_b^0(z) =z </math> | : <math> \exp_b^0(z) =z </math> | ||
: <math> \exp_b^1(z) =\exp_b(z) </math> | : <math> \exp_b^1(z) =\exp_b(z) </math> | ||
: <math> \exp_b^ | : <math> \exp_b^2(z) =\exp_b\!\Big (\exp_b(z)\Big) </math> | ||
: <math> \exp_b^{c+1}(z) =\exp_b(\exp_b^c(z) </math> | : <math> \exp_b^{c+1}(z) =\exp_b\!\Big(\exp_b^c(z)\Big) </math> | ||
For non-integer values of <math>c</math>, the iterated exponential can be defined as | For non-integer values of <math>c</math>, the iterated exponential can be defined as | ||
: <math> \exp_b^c(z) = | : <math> \exp_b^c(z) = | ||
\mathrm{sexp}_b\Big(c+ | \mathrm{sexp}_b\!\Big(c+ | ||
{\mathrm{sexp}_b}^{-1}(z)\Big) </math> | {\mathrm{sexp}_b}^{-1}(z)\Big) </math> | ||
where <math> \mathrm{sexp}_b(z) </math> is function <math>F</math> | where <math> \mathrm{sexp}_b(z) </math> is function <math>F</math> satisfying conditions | ||
:<math>F(z+1)=\exp_b(F(z))</math> | :<math>F(z+1)=\exp_b\!\Big(F(z)\Big)</math> | ||
:<math>F(0)=1</math> | :<math>F(0)=1</math> | ||
:<math>F(z)~ \mathrm{ ~is~ holomorphic~ and~ bounded~ | :<math>F(z)~ \mathrm{ ~is~ holomorphic~ and~ bounded~ at}~ |\Re(z)|<1</math> | ||
The inverse function is defined with condition | The inverse function is defined with condition | ||
: <math>F\Big(F^{-1}(z)\Big)=z</math> | : <math>F\Big(F^{-1}(z)\Big)=z</math> | ||
and, within some range of values of <math>z</math> | and, within some range of values of <math>z</math> | ||
: <math>F^{-1}\Big (F(z)\Big)=z</math> | : <math>F^{-1}\Big(F(z)\Big)=z</math> | ||
If in the notation <math>\exp_b^c</math> the superscript is omitted, it is assumed to be unity; for example | |||
<math>\exp_b^1=\exp_b</math>. If the subscript is omitted, it is assumed to be <math>\mathrm{e}</math>, id est, <math>\exp^c=\exp_\mathrm{e}^c</math> | |||
Function <math>f=\exp^c(z)</math> is shown in figure with levels of constant real part and levels of constant imaginary part. Levels | |||
<math>\Re(f)=-3,-2,-1,0,1,2,3,4,5,6,7,8,9</math> and | |||
<math>\Im(f)=-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14</math> 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 <math>\Re(f)=-0.2, -0.4, -0.6, -0.8</math> are shown with thin red lines. | |||
Levels <math>\Im(f)= 0.2, 0.4, 0.6, 0.8</math> are shown with thin green lines. | |||
Levels <math>\Re(f)=\Re(L)</math> and | |||
Levels <math>\Im(f)=\Im(L)</math> are marked with thick green lines, where <math>L\approx 0.31813150520476413 +1.3372357014306895~ \mathrm{i}</math> is [[fixed point]] of logarithm. | |||
At non-integer values of <math>c</math>, <math>L</math> and <math>L^*</math> are [[branch point]]s of function <math>\exp^c</math>; in figure, the cut is placed parallel to the real axis. At <math>c<0</math> 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 <math> y=\exp^c(x)</math> is ploted in figure versus <math>x</math> for values | |||
<math>c=0,\pm 0.1, \pm 0.5, \pm 0.9, \pm 1, \pm 2</math>.<br> | |||
in [[programming languages]], inverse function of exp is called [[log]]. | |||
For [[logarithm]] on base e, notation ln is also used. In particular, | |||
<math>\exp^{-1}(x)=\ln(x)</math>, | |||
<math>\exp^{-2}(x)=\ln\big(\ln(x)\big)</math> and so on. | |||
==References== | |||
{{reflist}} | |||
* Ahlfors, Lars V. (1953). Complex analysis. McGraw-Hill Book Company, Inc.. | |||
* H.Kneser. ``Reelle analytische Losungen der Gleichung <math>\varphi(\varphi(x))=\mathrm{e}^{x}</math> und verwandter Funktionalgleichungen''. Journal fur die reine und angewandte Mathematik, <b> 187</b> (1950), 56-67. | |||
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* H.Kneser. ``Reelle analytische L\"osungen der Gleichung <math>\varphi(\varphi(x))=\mathrm{e}^{x}</math> und verwandter Funktionalgleichungen''. Journal f\"ur die reine und angewandte Mathematik, <b> 187</b> (1950), 56-67. | |||
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Latest revision as of 17:00, 14 August 2024
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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
- 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 :
The exponential is related to the trigonometric functions sine and cosine by de Moivre's formula:
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 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
- ↑ 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.