Basic expounts A — Wikipedia

Posted on May 28, 2021 by lordneo

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In real analysis, the basic exponential $a$ is the noted function $exp a$ who, to all real $x$ , associates the real $a x$ . She only makes sense for a real $a$ strictly positive. It extends to all the reals the function, defined on all natural integers, which to the whole $n$ associated $a n$ . It is therefore the continuous version of a geometric suite.

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It is expressed using the exponential usual functions and the nepalian logarithm in the form

{displaystyle exp _{a}(x)=a^{x}={rm {e}}^{xln(a)}.}

It can be defined as the only continuous function on ℝ, taking the value $a$ in $first$ and transforming a sum into a product.

For $a$ different from $first$ , this is the reciprocal of the basic logarithm function $a$ . These functions are sometimes called anti -data functions. The case $a = and$ corresponds to the exponential functions and Nepérian logarithm.

Exponential functions are the only derivable functions on ℝ, proportional to their derivative and taking the value $first$ in $0$ . They make it possible to model the physical or biological phenomena in which the speed of growth is proportional to the size of the population.

We also find the term exponential functions for functions whose expression is $N a x$ .

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We consider a real $a$ strictly positive; It is easy to define $a n$ like the product of $a$ by himself $n$ times for any whole $n$ greater than or equal to $first$ ,

{displaystyle exp _{a}(n)=a^{n}={underset {n{text{ fois}}}{underbrace {atimes atimes cdots times a} }}}

then define $a 0 = 1$ And $a -n = first / a n$ . We easily demonstrate the property $a n + m = a n \times a m$ . This fairly natural construction corresponds to the so -called growth or exponential decrease phenomena.

Example 1: Imagine a population whose size increases by 30% every 10 years. If we note $N$ the population in 1900, it is easy to calculate the population in 1910, 1920 … which will be $N \times 1.3$ , Then $N \times 1.3 2$ … To reach the end of n decades $N \times 1.3 n$ . It is even possible to determine the population in 1890, 1880 … which will be $N \times 1.3 -1$ , $N \times 1.3 -2$ …
Example 2: Carbon 14 has a radioactive period of period $T = 5 730$ years which means that all $T$ years, the number of radioactive particles has been divided by 2. If we measure, at a given moment, the number $N$ radioactive particles, after $n$ periods, the number of radioactive particles is only $N \times (1/2) n$ .

The question that arises is to determine the size of the population or the number of radioactive particles between two measures (the decade for the population or the period for the particle). It is therefore a question of “filling the holes between the whole”. An attempt can be made thanks to the root $n$ -th: if the population has multiplied in 10 years by 1.3, we are trying to determine by how multiplied it is each year. It is multiplied by a real $q$ such as $q ten = 1.3$ , that’s to say $q = ten \sqrt 1.3$ that we note $1.3 1/10$ .

We are therefore able to define $a r$ For non -whole exhibitors:

{displaystyle exp _{a}(1/q)=a^{1/q}={sqrt[{q}]{a}}}

{displaystyle exp _{a}(p/q)=a^{p/q}=({sqrt[{q}]{a}})^{p}={sqrt[{q}]{a^{p}}}}

We have “filled the holes” and defined $a r$ for everything $r$ rational. To define $a x$ for all real $x$ , we must add an argument of continuity: all real $x$ is “as close as we want” of a rational $p / q$ ; the value of $a x$ will then be “close to” $a p / q$ .

This intuitive idea of what could be $a x$ appears very early-at the same time as the exponential notation, that is to say from the XVII ^{It is}century ^{[ first ]}. But it will be necessary to wait for the following centuries to see in $x \mapsto a x$ :

a function;
verifying $a x + y = a x a and$ , that is to say transforming a sum into a product;
continue ;
reciprocal of a logarithm function (which transforms a product in short);
Derivable and whose derivative is proportional to the function.

There are several possible entry points for the definition of exponential function: by its algebraic properties (transforms a sum into a product), by the property of its derivative (derivative proportional to the function), or by its relations with the exponential function and the Neper Logarithm function.

Table of Contents

By algebraic property [ modifier | Modifier and code ]

Definition – We call real exponential function, any function of R In R , not identically zero and continuous in at least one point, transforming a sum into a product, that is to say checking the functional equation

{displaystyle forall u,vin mathbb {R} quad f(u+v)=f(u)times f(v).}

Such a function $f$ is continuous and strictly positive and for any real $a > 0$ , the special one $f$ such as $f (1) = a$ is called basic exponential $a$ and is noted $exp a$ .

In other words: these functions are the continuous morphisms of ( R , +) in ( R ₊*, ×), and are in bijection with R ₊* via $f \mapsto f (first)$ .

The relationship

{displaystyle f(u)=fleft(2{frac {u}{2}}right)=left[fleft({frac {u}{2}}right)right]^{2}}

ensures that the function is with positive values.

The functional equation also guarantees that all these values are not zero as soon as one of them is.

Then, considerations similar to those developed in the previous section ensure the existence ^{[ 2 ]}and uniqueness ^{[ 3 ]}, for any real $a > 0$ , of a function $f$ defined on rational , checking the functional equation, and taking in 1 the value $a$ .

We demonstrate ^{[ 3 ]}continuity and – by density of ℚ in ℝ – The uniqueness of a function verifying the functional equation, taking in $first$ the value $a$ , and continues in at least one point. Its existence is obtained by extension by continuity:

Details

It is easy to check that the function $x \mapsto a x$ (defined for the moment only on rational) is monotonous and that the following $a 1/2 n$ Converges to 1. This, attached to the functional equation, makes it possible to show that the function is Cauchy-Continue on ℚ, therefore extendable by continuity to ℝ. By continuity and density, this extension to ℝ still verifies the functional equation.

We can notice that – apart from the constant function $first$ , that corresponds to $a = 1$ – All these applications $f : ℝ \to ]0, +\infty[$ are bijective. So these are isomorphisms of ( R , +) in ( R ₊*, ×).

We prove that then $f$ is differential and checks the differential equation:

{displaystyle f'(x)=f'(0)times f(x)quad {text{et}}quad f(0)=1.}

Demonstrations

first ^remethod. -According to § “by a differential equation” below, the functions $g k : x \mapsto exp(kx)$ check $g’ k = kg k$ , $g k (0) = 1$ And transform the sums into products. For $k = exp -1 (a)$ , on a $g k (1) = a$ therefore by uniqueness, $g k = f$ .

2 ^{It is}method. – To demonstrate that a continuous function transforming a sum into a product is necessarily derivable, we can rely on the fact that a continuous function has primitives ^{[ 4 ]}. If we note $F$ a primitive of $f$ , we can write

{displaystyle int _{0}^{1}f(x)f(t)mathrm {d} t=f(x)int _{0}^{1}f(t)mathrm {d} t=f(x)(F(1)-F(0))}

but also

{displaystyle int _{0}^{1}f(x)f(t)mathrm {d} t=int _{0}^{1}f(x+t)mathrm {d} t=F(x+1)-F(x).}

Function $f$ being strictly positive, $F$ is strictly increasing and $F (first) - F (0)$ is then not zero. By confronting the two equalities, we can write

{displaystyle f(x)={frac {F(x+1)-F(x)}{F(1)-F(0)}},}

which proves that, $f$ speaking as a linear combination of derivable functions, $f$ is differentiable.

By deriving equality

{displaystyle f(x+u)=f(x)f(u)}

compared to $x$ , we obtain

{displaystyle f'(x+u)=f'(x)f(u)}

Then, taking $x$ equal to $0$ ,

{displaystyle f'(u)=f'(0)f(u).}

Using the exponential function and the Neper Logarithm function [ modifier | Modifier and code ]

Definition – Either $a$ A strictly positive real. We call basic exponential function $a$ the function defined on ℝ by

{displaystyle f(x)={rm {e}}^{xln(a)},}

Or $x \mapsto and x$ is the exponential function and $ln$ The Neper Logarithm function.

This function is very continuous, transforms a sum into a product and takes the value $a$ in 1.

By a differential equation [ modifier | Modifier and code ]

Definition – An exponential function is called any derivable function verifying the differential equation and the following initial condition:

{displaystyle f’=kfquad {text{et}}quad f(0)=1}

Or $k$ is real.

We can notice that for such a function, $k$ is the value of the derivative in 0.

Assuming only known the existence of a solution to $k = 1$ (function $exp$ ), an obvious solution for $k$ any function $x \mapsto exp(kx)$ .

We show ^{[ 5 ]}that this solution is the only one. In addition, the solution transforms any amount into a product ^{[ 6 ]}, therefore its definition coincides with that above “by algebraic property”, to $a = exp(k)$ .

As a reciprocal of the logarithm functions [ modifier | Modifier and code ]

Definition – Either $a$ A strictly positive real, different from $first$ . The basic logarithm function $a$ is a bijection of R* ₊In R . We call basic exponential function $a$ His reciprocal bijection:

{displaystyle {bigl (}xin mathbb {R} ,quad y=exp _{a}(x){bigr )}Leftrightarrow {bigl (}yin left]0,+infty right[,quad x=log _{a}(y){bigr )}.}

The logarithm function being continuous, transforming a product in short and taking the value $first$ in $a$ , its reciprocal bijection is continuous, transforms a sum into a product and takes the value $a$ in $first$ .

Algebraic properties [ modifier | Modifier and code ]

For all strictly positive real $a$ And $b$ and for all real $x$ And $and$ :

${displaystyle a^{x}=exp(xln(a))={rm {e}}^{xln(a)},quad a^{x}=b^{xlog _{b}(a)},quad a^{xy}=left(a^{x}right)^{y}.}$
Applications $exp a : x \mapsto a x$ are morphisms of groups (abéliens) of ( R , +) in ( R ₊*, ×):

${displaystyle a^{x+y}=a^{x}a^{y},quad a^{0}=1,quad {frac {1}{a^{x}}}=a^{-x}.}$
These morphisms constitute an isomorphic group to ( R ₊*, ×) ( via $a \mapsto exp a$ ) – So also to ( R , +):

${displaystyle a^{x}b^{x}=(ab)^{x},quad 1^{x}=1,quad {frac {1}{a^{x}}}=left({frac {1}{a}}right)^{x},quad a^{1}=a.}$

Function study [ modifier | Modifier and code ]

The basic exponential function $a$ is indefinitely derivable on R and its derivative has the expression

{displaystyle exp _{a}'(x)=ln(a)exp _{a}(x).}

Since the exponential function is always positive, the sign of its derivative depends only on the sign of $ln(a)$ . The function is therefore strictly increasing when the base $a$ is strictly larger than $first$ ; it is strictly decreasing when the base is less than $first$ and constant if we took as a base $a = 1$ .

The limits of the basic exponential function $a$ depend on the position of $a$ compared to $first$ :

and $a > 1$ SO ${Displaystyle Lim _ {xto +Infty} a^{x} = +Infty quad {rm {et}} quad lim _ {xto -infty} a^{x} = 0 ~;}$
and $a <1$ SO ${Displaystyle Lim _ {xto +Infty} a^{x} = 0quad {rm {et}} lim _ {xto -infty} a^{x} = +Infty.}$

The exponential function has a predictable behavior compared to the power function: in the event of indeterminacy in $+\infty$ , it is the exponential that prevails:

For all real

a > 1

And

b

{displaystyle lim _{xto +infty }{frac {a^{x}}{x^{b}}}=+infty .}

It is both convex (therefore convex) and logarithmically concave (in) .

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Basic expounts A — Wikipedia

By algebraic property [ modifier | Modifier and code ]

Using the exponential function and the Neper Logarithm function [ modifier | Modifier and code ]

By a differential equation [ modifier | Modifier and code ]

As a reciprocal of the logarithm functions [ modifier | Modifier and code ]

Algebraic properties [ modifier | Modifier and code ]

Function study [ modifier | Modifier and code ]

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