Exponential

The exponential function is one of the most important application S in analyzes, or more generally in Mathématiques and in its scopes of application. There exist several equivalent definitions: a continuous morphism of groups R → R ^* or C → C ^*, a solution of a linear differential equation of order a, or a analytical Function with a real variable or complex nap of a whole Series. These three definitions make it possible to extend the definition to the Géométrie riemannienne, the theory of the groups of Dregs, or to the study of the algebras of Banach.

Without extending on these generalizations, the properties of the exponential application are largely accessible. Its elementary applications relate to the resolution of the differential equations, the installation of the theory of Fourier, the study of the growth of the groups, etc

Approaches popularized

If has is a real number and N is an integer, then the “exponential one of N in base has ” is equal to “ has power N ” is:

$\backslash \; exp\_a\; (N)\; =\; a^n\; =\; \backslash \; underset\; \{N\; \backslash \; text\; \{time\}\}\; \{\backslash \; underbrace\; \{has\; \backslash \; times\; has\; \backslash \; times\; \backslash \; cdots\; \backslash \; times\; has\}\}$

One can extend this function to the nonwhole numbers. It is shown whereas the exponential ones are the reciprocal functions of the Logarithme S log _has , and in addition that the goniometrical functions can be expressed in a simple way with the exponential ones.

These functions are derived and are integrated in a very simple way, and intervene in many solutions of differential equations.

There exists a base E such as exponential the basic E is the reciprocal function Napierian logarithm ln. In this base, the Dérivée from the exponential function is equal to itself is $(e^x)\; \text{'}=\; e^x$. It is this base which is used, and it is with it which one generally refers if one does not specify another of them.

Definitions

Continuous Morphism

It is natural to want to seek to describe the continuous morphisms F : R → R ^* (or by analogy C → C ^*). In other words, one seeks the continuous applications F checking the functional equation following:

$f\; (u+v)\; =f\; (U)\; \backslash \; cdot\; F\; (v)$. Necessarily, F is derivable and checks the differential equation: $f\text{'}\; (X)\; =f\text{'}\; (0)\; \backslash \; cdot\; F\; (X)$. By imposing f' (0) =1, one determines F with a multiplicative constant 1. The functional equation imposes however F (0) =1, therefore F is only given.

More generally, for a topological group G , one calls sub-group with a parameter any continuous morphism R → G . Certain works can replace the assumption of continuity by the mesurability for example.

See also: Sub-group with a parameter

Differential equation

The second definition, equivalent to the preceding one, is the single derivable application F : R → R ^* checking the differential equation:

$f\text{'}\; (X)\; =f\; (X)$ with like initial condition $f\; (0)\; =1$.

This definition spreads for the groups of Dregs and the geodetic ones in the riemanniennes varieties.

See also: Exponential of a group of Exponential Dregs, of a variety riemannienne

Series

Lastly, by applying the method of research solution analytical of the linear differential equations, one can define the exponential application $\backslash \; exp$ or $x\; \backslash \; mapsto\; e^x$ as the sum of a whole series of infinite ray of convergence:

$\backslash \; exp\; (X)\; =\; \backslash \; sum\_\; \{N\; =\; 0\}\; ^\; \{+\; \backslash \; infty\}\; \{x^n\; \backslash \; over\; N!\}$, where $n!$ is the Factorielle of $n$.

This last approach allows an immediate extension of the definition of exponential the algebras of Banach.

See also: Algebra of Banach

Properties

Real exponential functions

Because of continuity, supposed in the three definitions given, if X is real, then exp ( X ) is a strictly positive reality. In addition the function exp of $\backslash \; R$ in $\backslash \; R\_+^*$ is strictly increasing, continues, continuously derivable, infinitely derivable, and still better analytical (developable IE in whole series in the vicinity of any point). Moreover, $\backslash \; lim\_\; \{X\; \backslash \; to\; -\; \backslash \; infty\}\; \backslash \; exp\; (X)\; =0$ and $\backslash \; lim\_\; \{X\; \backslash \; to\; +\; \backslash \; infty\}\; \backslash \; exp\; (X)\; =+\; \backslash \; infty$, it thus admits a reciprocal Application, which is the function Napierian logarithm ln , definite on $\backslash \; R\_+^*$.

As the derivative successive of exp are exp , the derivative second is positive. Thus exp is convex.

By using the function Napierian logarithm ln , one can define for all $a\; >\; 0$ the exponential function basic has noted $\backslash \; exp\_a$ or $x\; \backslash \; mapsto\; a^x$, by:

$\backslash \; forall\; X,\; \backslash \; a^x\; =\; \backslash \; exp\; (\backslash \; ln\; (a)\; X)$.

The exponential function also makes it possible to define the goniometrical functions with the formulas of Euler and the functions hyperbolic. Thus we see that all the elementary functions, except for the polynomial functions, are expressed starting from the exponential function, in a form or another.

The exponential functions “transform an addition into a multiplication”, as these properties show it:

$a^0\; =\; 1$

$a^1\; =\; a$

$a^\; \{X\; +\; there\}\; =\; a^x\; \backslash \; cdot\; a^y$

$a^\; \{X\; there\}\; =\; \backslash \; left\; (a^x\; \backslash \; right)\; ^y$

$\backslash \; frac1\; \{a^x\}\; =\; \backslash \; left\; (\backslash \; frac1a\; \backslash \; right)\; ^x\; =\; a^\; \{-\; X\}$

$a^x\; b^x\; =\; (has\; b)^x$

They are valid for all strictly positive realities has and B and for all realities X and there .

For has real strictly positive, $\backslash \; exp\_a$ is the only monotonous morphism of the additive group $\backslash \; R$ in the multiplicative group $\backslash \; R\_+^*$ of strictly positive realities checking $\backslash \; exp\_a\; (1)\; =a$.

For a=1 , the exponential function is constant and equal to 1, and bijective is not thus any more . When has ≠ 1 , the exponential function is an isomorphism of the additive group $\backslash \; R$ on the multiplicative group $\backslash \; R\_+^*$; strictly growing if a>1 and strictly decreasing if a<1 .

In addition, it is possible to write expressions utilizing quotients or roots by using the exponential notation. For example:

$\backslash \; frac1a\; =\; a^\; \{-\; 1\}$

$\backslash \; sqrt\; \{has\}\; =\; a^\; \{1/2\}$

$\backslash \; sqrt\; \{has\}\; =\; a^\; \{1/n\}$

Exponential function in the complex plan

One can define the exponential function complexes in 2 ways:

1. $\backslash \; exp\; (ix)\; =\; \backslash \; cos\; (X)\; +i\; \backslash \; sin\; (X)$

2. By using the development in series of exponential which makes it possible to extend this one to the complex plan.
3. : $\backslash \; exp\; (Z)\; =\; \backslash \; sum\_\; \{N\; =\; 0\}\; ^\; \{\backslash \; infty\}\; \{z^n\; \backslash \; over\; N!\}$

The exponential function checks the following important properties then, for all Z and W :

$\backslash \; exp\; (Z\; +\; W)\; =\; \backslash \; exp\; (Z)\; \backslash \; exp\; (W)$

$\backslash \; exp\; (0)\; =\; 1$

$\backslash \; exp\; (Z)\; \backslash \; 0$

$\backslash \; exp\; \text{'}(Z)\; =\; \backslash \; exp\; (Z)\; \backslash !$

These formulas are shown using the formulas of Trigonométrie or using the concept of Produit of Cauchy of two series according to the mode of definition of the exponential one.

The exponential function in the complex plan is a holomorphic Fonction which is periodic, of imaginary period $2\; \backslash \; pi\; i$ and checks:

$\backslash \; exp\; (+\; Bi\; has)\; =\; \backslash \; exp\; (a)\; \backslash \; cdot\; (\backslash \; cos\; (b)\; +\; I\; \backslash \; sin\; (b))$

where $a$ and $b$ are real numbers. This formula is the bond between the exponential function and the trigonometrical functions , and this is why to prolong the natural logarithm with the whole of the complex numbers, gives a multiform function naturally $Z\; \backslash \; mapsto\; \backslash \; ln\; (Z)$, called Logarithme complexes.

One can define exponential the more general:

for all complex numbers Z and W , $z^w\; =\; \backslash \; exp\; (\backslash \; ln\; (Z)\; W)$

It is also a multiform function. The properties above of exponential remain true with the proviso of suitably interpreting them like relations between multiform functions.

Application

Goniometrical function

See also: goniometrical Function

The exponential function is of a capital utility in trigonometry. Thanks to the Formules of Euler (which one shows starting from the definition $\backslash \; exp\; (iz)\; =\; \backslash \; cos\; (Z)\; +i\; \backslash \; sin\; (Z)$) gives us a direct link between the function cosine and sines , real or not, and the complex exponential function.

$\backslash \; cos\; X\; =\; \{e^\; \{ix\}\; +\; e^\; \{-\; ix\}\; \backslash \; over\; 2\}$

$\backslash \; sin\; X\; =\; \{e^\; \{ix\}\; -\; e^\; \{-\; ix\}\; \backslash \; over\; 2i\}$

These formulas make it possible to find the majority of the trigonometrical Formules, in particular

$\backslash \; cos\; (a+b)=\; \backslash \; cos\; (a)\; \backslash \; cos\; (b)\; -\; \backslash \; sin\; (a)\; \backslash \; sin\; (b)\; ~$

$\backslash \; sin\; (a+b)=\; \backslash \; sin\; (a)\; \backslash \; cos\; (b)\; +\; \backslash \; sin\; (b)\; \backslash \; cos\; (a)\; ~$

from which one can find almost all the others.

The exponential function is also a means easy (although calculations can be long) to linearize goniometrical functions.

$\backslash \; cos^\; \{N\}\; X\; =\; \backslash \; left\; (\backslash \; frac\; \{e^\; \{ix\}\; +e^\; \{-\; ix\}\}\; \{2\}\; \backslash \; right)\; ^\; \{N\}$

$\backslash \; sin^\; \{N\}\; x=\; \backslash \; left\; (\backslash \; frac\; \{e^\; \{ix\}\; -\; e^\; \{-\; ix\}\}\; \{2i\}\; \backslash \; right)\; ^\; \{N\}$

It is then enough to develop the sum thanks to the Formule of the binomial theorem, to gather the terms knowing that

$e^\; \{I\; (n-k)\; X\}\; e^\; \{-\; ikx\}\; =\; e^\; \{I\; (n-2k)\; X\}$

$e^\; \{imx\}\; +\; e^\; \{-\; imx\}\; =\; 2\; \backslash \; cos\; (MX)$

$e^\; \{imx\}\; -\; e^\; \{-\; imx\}\; =\; 2i\; \backslash \; sin\; (MX)$

The exponential function finds also its utility when one wants to show the Formule of Moivre.

Exponential function and hyperbolic trigonometry

From the exponential function, one can define the functions of hyperbolic trigonometry, defining the hyperbolic functions hyperbolic Cosinus, CH (or English cosh) and hyperbolic Sinus, HS (or English sinh), used partly in the resolutions of the differential equations of second order.

Theory of Fourier

See also: Theory of Fourier

Linear differential equation

See also: linear differential equation

The major importance of the exponential functions in sciences, comes owing to the fact that it are constant multiples of their clean derived. has being a real number or complex, one a:

$(\backslash \; lambda\; e^\; \{ax\})\; \text{'}=\; has\; \backslash \; lambda\; e^\; \{ax\}$

or more exactly, there is $\backslash \; varphi:\; X\; \backslash \; mapsto\; \backslash \; lambda\; e^\; \{ax\}$ if and only if

$\backslash \; varphi\text{'}\; =\; has\; \backslash \; varphi$ and $\backslash \; varphi\; (0)\; =\; \backslash \; lambda$

If a size grows or decrease, according to time and that the speed of “its race” is proportional to “its size”, as in the case of the growth of a population, continuous made up interests or radioactive decrease, then this size can be expressed like constant once an exponential function of time.

The exponential function basic E is solution of the differential equation elementary:

$y\text{'}\; =\; y$

and one frequently meets it in the solutions of differential equations. In particular, the solutions of a linear differential equation can be written using the exponential functions. One also finds them in the solutions of the differential equations of Schrödinger, Laplace or in the differential equation of the simple harmonic Mouvement.

Growth of the groups

See also: Growth of the groups

See too

• Logarithm
• Differential equation
• Exponential Trigonometry
• of a matrix
• Number E

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