Exponential function

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The natural exponential function and = e^x

In mathematics, one exponential function is a function of form ${displaystyle f(x)=ab^{x}}$ in which argument x is presented as an exponent. A function of form ${displaystyle f(x)=ab^{cx+d}}$ is also an exponential function, since it can be rewritten as:

{displaystyle ab^{cx+d}=left(ab^{d}right)left(b^{c}right)^{x}.}

As functions of a real variable, exponential functions are characterized only by the fact that the growth rate of such function (i.e. its derivative) is directly proportional to the value of the function. The constant proportionality of this relationship is the natural logarithm of the base b: ${displaystyle {frac {d}{dx}}b^{x}=b^{x}log _{e}b.}$ The constant e = 2.71828... is the only basis for which the proportionality constant is 1, so that the derivative of the function is in itself: ${displaystyle {frac {d}{dx}}e^{x}=e^{x}log _{e}e=e^{x}.}$ Since the change of the base of the exponential function simply results in the emergence of an additional constant factor, it is computationally convenient to reduce the study of exponential functions in mathematical analysis to the study of this particular function, conventionally called the "natural exponential function", or simply, "the exponential function" and denoted by ${displaystyle xmapsto e^{x}}$ or ${displaystyle xmapsto exp(x).}$ While both notations are common, the first is usually used for the simplest exponents, while the latter tends to be used when the exponent is a complicated expression.

The exponential function satisfies the fundamental multiplier identity ${displaystyle e^{x+y}=e^{x}e^{y},}$ for everything ${displaystyle x,yin mathbb {R}.}$ This identity extends to the exponents of complex values. It can be shown that each continuous solution, different from zero, of the functional equation ${displaystyle f(x+y)=f(x)f(y)}$ is an exponential function, ${displaystyle f:mathbb {R} to mathbb {R} xmapsto b^{x},}$ with the fundamental multipliative identity, together with the definition of the number $e$ Like $e 1$ , shows that ${displaystyle e^{n}=underbrace {etimes cdots times e} _{n{text{ términos}}}}$ for positive integers $n$ and relates the exponential function to the elemental notion of exponentiation.

The argument of the exponential function can be any real or complex number, or even an entirely different type of mathematical object (for example, a matrix).

Its ubiquitous appearance in pure and applied mathematics has led mathematician W. Rudin to believe that the exponential function is "the most important function in mathematics". With applied adjustments, exponential functions model a relationship in which a constant change in the independent variable provides the same proportional change (that is, percentage increase or decrease) in the dependent variable. This occurs widely in the natural and social sciences; therefore, the exponential function also appears in a variety of contexts within physics, chemistry, engineering, mathematical biology, and economics.

The chart of the ${displaystyle y=e^{x}}$ is tilted upwards, and increases faster as $x$ increase. The chart is always above the axis $x$ but may be arbitrarily close to him for $x$ negative; thus, the axis $x$ It's a horizontal asymptote. The slope of the tangent to the graph at each point is equal to its coordinate and at that point, as indicated by its derivative function. Its reverse function is the natural logarithm, denoted ${displaystyle log}$ ${displaystyle ln}$ or ${displaystyle log _{e};}$ because of this, some ancient texts refer to the exponential function as the antilogaritmo.

Formal definition

The exponential function (in blue) and the sum of the first n + 1 terms of their power series (in red).

The actual exponential function ${displaystyle exp:mathbb {R} to mathbb {R} }$ can be characterized in several equivalent ways. More commonly, it is defined by the following series of powers:

{displaystyle exp(x)=sum _{k=0}^{infty }{frac {x^{k}}{k!}}=1+x+{frac {x^{2}}{2}}+{frac {x^{3}}{6}}+{frac {x^{4}}{24}}+cdots }

As the convergence radius of this series of powers is infinite, this definition is, in fact, applicable to all complex numbers. ${displaystyle zin mathbb {C} }$ .

The term-to-end differentiation of this series of powers reveals that ${displaystyle (d/dx)(exp x)=exp x}$ for all $x$ which leads to another common characterization of $exp(x)$ as the only solution of differential equation

{displaystyle y'(x)=y(x),}

satisfying the initial condition ${displaystyle y(0)=1.}$

Based on this characterization, the string rule shows that its reverse function, the natural logarithm, satisfies ${displaystyle (d/dy)(log _{e}y)=1/y}$ for $0,}" xmlns="http://www.w3.org/1998/Math/MathML">and▪0,{displaystyle and censorship0,}0,}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc81091cdfd280fa156bb36c35bbbdac764a2b43" style="vertical-align: -0.671ex; width:6.063ex; height:2.509ex;"/>$ or ${textstyle log _{e}y=int _{1}^{y}{frac {1}{t}},dt.}$ This relationship leads to a less common definition of actual exponential function $exp(x)$ as the solution $y$ to the equation

{displaystyle x=int _{1}^{y}{frac {1}{t}},dt.}

By means of the binomial theorem and the definition of the power series, the exponential function can also be defined as the following limit:

{displaystyle e^{x}=lim _{nto infty }left(1+{frac {x}{n}}right)^{n}.}

Overview

The red curve is the exponential function. The black horizontal lines show where it crosses the green vertical lines.

The exponential function arises when a quantity rises or falls at a rate proportional to its present value. One such situation is continuously compound interest, and in fact, it was this observation that led Jacob Bernoulli in 1683 to the number

{displaystyle lim _{nto infty }left(1+{frac {1}{n}}right)^{n}}

now known as e. Later, in 1697, Johann Bernoulli studied the calculus of the exponential function.

If a principal amount of 1 earns interest at an annual rate of $x$ monthly compounding, then the interest earned each month is $x / 12$ times the current value, so each month the total value is multiplied by $(1 + x / 12)$ , and the value at the end of the year is $(1 + x / 12) 12$ . If, instead, the interest worsens daily, this becomes $(1 + x / 365) 365$ . Letting the number of time intervals per year grow without limit leads to the limit definition of the exponential function,

{displaystyle exp(x)=lim _{nto infty }left(1+{frac {x}{n}}right)^{n}}

first given by Leonhard Euler. This is one of several characterizations of the exponential function; Others involve series or differential equations.

From any of these definitions it can be shown that the exponential function obeys the basic exponentiation identity,

{displaystyle exp(x+y)=exp(x)cdot exp(y)}

which justifies the $e x$ notation.

The derivative (rate of change) of the exponential function is the exponential function itself. More generally, a function with a rate of change proportional to (rather than equal to) the function itself is expressible in terms of the exponential function. This property of function leads to exponential growth or exponential decay.

The exponential function is extended to a complete function in the complex plane. Euler's formula relates its values in purely imaginary arguments to trigonometric functions. The exponential function also has analogs for which the argument is a matrix, or even an element of a Banach algebra or a Lie algebra.

Derivatives and differential equations

The derivative of the exponential function is equal to the value of the function. From any point P in the curve (blue), draw a tangent line (red) and a vertical line (green) with height h, forming a rectangle triangle with a base b on the axis x. Since the slope of the red tangent line (the derivative) in P is equal to the relationship between the height of the triangle and the base of the triangle (increase on the execution), and the derivative is equal to the value of the function, h must be equal to the relationship h a b. Therefore, the base b It must always be 1.

The importance of the exponential function in mathematics and science comes mainly from its definition as a unique function that is equal to its derivative and is equal to 1 when $x = 0$ . That is to say,

{displaystyle {frac {d}{dx}}e^{x}=e^{x}quad {text{y}}quad e^{0}=1.}

Functions of the form $ce x$ for the constant $c$ are the only functions that are equal to their derivative (by the Picard-Lindelöf theorem). Other ways of saying the same thing include:

The slope of the chart at any point is the height of the function at that point.
The rate of increase in function $x$ equals the value of the function in $x$ .
Function solves differential equation $and ♫ and$ .
$Exp$ is a fixed point of derivation as functional.

If the rate of growth or decay of a variable is proportional to its size, as is the case with unlimited population growth (see Malthusian catastrophe), continuously compounded interest, or radioactive decay, then the variable can be written as an exponential function by time. Explicitly for any real constant k, a function $f : R \to R$ satisfies $f' = kf$ if and only if f (x) = cekx for some constant c. $k$ , a function satisfies if and only if $f (x) = ce kx$ for some constant $c$ .

Furthermore, for any differentiable function f(x), we find, by the chain rule:

{displaystyle {frac {d}{dx}}e^{f(x)}=f'(x)e^{f(x)}.}

Continued Fractions for ex

A continued fraction for $e x$ can be obtained through an identity of Euler:

{displaystyle e^{x}=1+{cfrac {x}{1-{cfrac {x}{x+2-{cfrac {2x}{x+3-{cfrac {3x}{x+4-ddots }}}}}}}}}

The following generalized continued fraction for $e z$ converges more quickly:

{displaystyle e^{z}=1+{cfrac {2z}{2-z+{cfrac {z^{2}}{6+{cfrac {z^{2}}{10+{cfrac {z^{2}}{14+ddots }}}}}}}}}

or, applying the substitution. $z = x / y$ :

{displaystyle e^{frac {x}{y}}=1+{cfrac {2x}{2y-x+{cfrac {x^{2}}{6y+{cfrac {x^{2}}{10y+{cfrac {x^{2}}{14y+ddots }}}}}}}}}

with a special case for $z = 2$ :

{displaystyle e^{2}=1+{cfrac {4}{0+{cfrac {2^{2}}{6+{cfrac {2^{2}}{10+{cfrac {2^{2}}{14+ddots ,}}}}}}}}=7+{cfrac {2}{5+{cfrac {1}{7+{cfrac {1}{9+{cfrac {1}{11+ddots ,}}}}}}}}}

This formula also converges, albeit more slowly, for z> 2. For example:

{displaystyle e^{3}=1+{cfrac {6}{-1+{cfrac {3^{2}}{6+{cfrac {3^{2}}{10+{cfrac {3^{2}}{14+ddots ,}}}}}}}}=13+{cfrac {54}{7+{cfrac {9}{14+{cfrac {9}{18+{cfrac {9}{22+ddots ,}}}}}}}}}

Complex shot

Exponential function in the complex plane. The transition from dark to clear colors shows that the magnitude of the exponential function is increasing to the right. Periodic horizontal bands indicate that the exponential function is periodic in the imaginary part of its argument.

As in the real case, the exponential function can be defined in the complex plane in several equivalent forms. The most common definition of the complex exponential function parallels the definition of the power series for real arguments, where the real variable is replaced by a complex one:

{displaystyle exp(z):=sum _{k=0}^{infty }{frac {z^{k}}{k!}}}

Multiplying two copies of these power series in the Cauchy sense, allowed by Mertens' theorem, shows that the defining multiplicative property of exponential functions remains valid for all complex arguments:

{displaystyle exp(w+z)=exp(w)exp(z)}

for everything

{displaystyle w,zin mathbb {C} }

The definition of the complex exponential function in turn leads to the appropriate definitions that extend the trigonometric functions to complex arguments.

In particular, when ${displaystyle z=it}$ ( $t$ the definition of the series produces the expansion

{displaystyle exp(it)={Big (}1-{frac {t^{2}}{2!}}+{frac {t^{4}}{4!}}-{frac {t^{6}}{6!}}+cdots {Big)}+i{Big (}t-{frac {t^{3}}{3!}}+{frac {t^{5}}{5!}}-{frac {t^{7}}{7!}}+cdots {Big)}}

In this expansion, the reorganization of the terms in real and imaginary parts is justified by the absolute convergence of the series. The real and imaginary parts of the previous expression in fact correspond to the expansions of the series ${displaystyle cos t}$ and ${displaystyle sin t}$ respectively.

This correspondence provides motivation to define the cosine and the breast for all complex arguments in terms of ${displaystyle exp(pm iz)}$ and the series of equivalent powers:

{displaystyle cos z:={frac {1}{2}}{Big [}exp(iz)+exp(-iz){Big ]}=sum _{k=0}^{infty }(-1)^{k}{frac {z^{2k}}{(2k)!}}}

and

{displaystyle operatorname {sen} z:={frac {1}{2i}}{Big [}exp(iz)-exp(-iz){Big ]}=sum _{k=0}^{infty }(-1)^{k}{frac {z^{2k+1}}{(2k+1)!}}}

for everything

{displaystyle zin mathbb {C} }

The functions exp, cos and without, thus defined, have an infinite radius of convergence by the test of relationship and, therefore, are complete functions (i.e., holomorfas in $mathbb{C}$ ). The range of the exponential function is ${displaystyle mathbb {C} setminus {0}}$ , while the ranges of complex sinus and cosine functions are $mathbb{C}$ in its entirety, according to the Picard theorem, which states that the range of a complete function is not constant $mathbb{C}$ or $mathbb{C}$ excluding a lacunary value.

These definitions for exponential and trigonometric functions trivially lead to Euler's formula:

{displaystyle exp(iz)=cos z+ioperatorname {sen} z}

for everything

{displaystyle zin mathbb {C} }

Alternatively, we could define the complex exponential function based on this relationship. Yeah. ${displaystyle z=x+iy}$ Where $x$ and $y$ They're real, we could define their exponential as

{displaystyle exp z=exp(x+iy):=(exp x)(cos y+ioperatorname {sen} y)}

where exp, cos and sin on the right hand side of the definition sign are to be interpreted as functions of a real variable, previously defined by other means.

Stop. ${displaystyle tin mathbb {R} }$ the relationship ${displaystyle {overline {exp(it)}}=exp(-it)}$ stays, so ${displaystyle |exp(it)|=1}$ for real $t$ and ${displaystyle tmapsto exp(it)}$ map the real line (mod) $2pi$ to the unitary circle. Based on the relationship between ${displaystyle exp(it)}$ and the unitary circle, it is easy to see that, restricted to real arguments, the previously given definitions of sin and cosine coincide with their most elementary definitions based on geometric notions.

Complex exponential function is periodic with the period $2pi i$ and ${displaystyle exp(z+2pi ik)=exp z}$ for all ${displaystyle zin mathbb {C}kin mathbb {Z} }$ .

When its domain extends from the real line to the complex plane, the exponential function retains the following properties:

${displaystyle e^{z+w}=e^{z}e^{w},}$
${displaystyle e^{0}=1,}$
${displaystyle e^{z}neq 0}$
${displaystyle {frac {mathrm {d} }{mathrm {d} z}}e^{z}=e^{z}}$
${displaystyle left(e^{z}right)^{n}=e^{nz},nin mathbb {Z} }$

Extending the natural logarithm to complex arguments produces the complex logarithm $log z$ , which is a multivalued function.

We can define a more general exponentiation:

{displaystyle z^{w}=e^{wlog z}}

for all complex numbers $z$ and w. This is also a multivalued function, even when $z$ is real. This distinction is problematic, since the multivalued functions $log z$ and $z w$ are easily confused with their single-valued equivalents when substituting a real number for $z$ . The rule about multiplying exponents for the case of positive real numbers must be modified in a multivalued context:

(e)^z) ^w ezw, but rather (e^z) ^w = E ^{(z + 2πin) w} multivalue on integers n

The exponential function maps any line in the complex plane to a logarithmic spiral in the complex plane with the center at the origin. Two special cases should be noted: when the original line is parallel to the real axis, the resulting spiral never closes on itself; when the original line is parallel to the imaginary axis, the resulting spiral is a circle of some radius.

3D graphics from the real part, the imaginary part and the exponential function module
$z = Re(e x + I)$
$z = Im(e x + I)$
$z = abs(e x + I)$

Considering the complex exponential function as a function involving four real variables:

{displaystyle v+iw=exp(x+iy)}

The graph of the exponential function is a two-dimensional surface that curves through four dimensions.

Starting with a color-coded part of the domain $xy$ , the following are representations of the chart as projected in a different way in two or three dimensions.

Charts of complex exponential function
Key:
$0:;{text{verde}}}" xmlns="http://www.w3.org/1998/Math/MathML">x▪0:Green{displaystyle x PHP0:;{text{verde}}}0:;{text{verde}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f029bc1f6d66cb5fe3739ca58e5d1addcaaa628" style="vertical-align: -0.338ex; width:13.669ex; height:2.176ex;"/>$
$<math alttext="{displaystyle xx.0:Red{displaystyle x vis0:;{text{rojo}}} <img alt="{displaystyle x$
$0:;{text{amarillo}}}" xmlns="http://www.w3.org/1998/Math/MathML">and▪0:yellow{displaystyle and censorship0:;{text{yellow}}}0:;{text{amarillo}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81d4ab8e682ddcd2eefec18b296dfed991bac921" style="vertical-align: -0.671ex; width:16.274ex; height:2.509ex;"/>$
$<math alttext="{displaystyle yand.0:blue{displaystyle and vis0:;{text{azul}}} <img alt="{displaystyle y$
Projection on the complex range plane (V/W). Compare to the following picture in perspective.
Projection in dimensions $x$ , $v$ and $w$ , producing a form of camped horn or funnel (conceived as a 2-D perspective image).
Projection in dimensions $y$ , $v$ and $w$ , producing a spiral form. ( $y$ range extended to ± 2π, again as image in perspective 2-D).

The second image shows how the complex domain plane is mapped to the complex range plane:

zero is assigned to 1
the real axis $x$ is assigned to the actual axis $v$ positive
the imaginary axis $y$ envelops around the unit circle at a constant angular speed
values with negative real parts are assigned within the unit circle
values with positive real parts are assigned outside the unit circle
values with a constant real part are assigned to zero-centred circles
values with a constant imaginary part are assigned to rays that extend from zero

The third and fourth images show how the graph in the second image is stretched in one of the other two dimensions not shown in the second image.

The third image shows the extended chart along the real axis $x$ . It shows that the chart is a surface of revolution on the axis $x$ of the graphic of the actual exponential function, which produces a form of horn or funnel.

The fourth image shows the graphic extended along the imaginary axis $y$ . Shows that the chart surface for values $y$ positive and negative really do not match the actual axis $v$ negative, but it forms a spiral surface around the axis $y$ . Because your values $y$ have spread to ± 2π, this image also represents the 2π periodicity in the imaginary value $y$ .

Figure of the actual part of an exponential function in the field of complexes

z=operatorname{Re} left (exp left(x + i y right)right)

Calculation of ab where both a and b are complex

Main article: Power

The complex exponentiation $a b$ can be defined by converting $a$ polar coordinates and using the identity $(e ln(a)) b = a b$ :

{displaystyle a^{b}=left(re^{theta i}right)^{b}=left(e^{ln(r)+theta i}right)^{b}=e^{left(ln(r)+theta iright)b}}

However, when $b$ is not an integer, this function is multivalued, because $θ$ is not unique.

General exponential function

If the complex number is taken as the basis a different from eand as variable the exponent z, you have to general exponential function w = f(z♪ ${displaystyle a^{z}}$ , is defined as:

${displaystyle w=a^{z}=e^{zoperatorname {Log} a}=e^{zln |a|}cdot e^{zioperatorname {Arg} a}}$

It is a family of univocal functions, not linked to each other, distinguished by the factors exp(2kπiz), where k is any integer.

Banach matrices and algebras

The definition of the power series of the exponential function makes sense for square matrices (for which the function is called an exponential matrix) and more generally in any algebra $B$ from Banach. In this configuration, $e 0 = 1$ , and $e x$ is invertible with e inverse $e - x$ for any x in $B$ . If $xy = yx$ , then $e x + y = e x e y$ , but this identity may fail to not switch $x$ and and.

Some alternative definitions lead to the same function. For example, $e x$ can be defined as:

{displaystyle lim _{nto infty }left(1+{frac {x}{n}}right)^{n}.}

O $e x$ can be defined as $f (1)$ , where $f : R \to B$ is the solution to the differential equation $f ' (t) = xf (t)$ with initial condition $f (0) = 1$ .

Lie Algebras

Given a Lie Group $G$ and its associated Lie algebra ${mathfrak {g}}$ , exponential map is a map ${mathfrak {g}}$ $G$ which satisfies similar properties. In fact, since $R$ is the Lie algebra of the Lie group of all the positive real numbers under multiplication, the ordinary exponential function for the actual arguments is a special case of the situation of the Lie algebra. Similarly, like the Lie group $GL(n, R)$ invertible matrices $n \times n$ It's like Lie algebra. $M(n, R)$ the space of all the matrices $n \times n$ , the exponential function for square matrices is a special case of exponential map of Lie algebra.

The identity $exp(x + y) = exp(x)exp(y)$ may fail for the elements of the Lie algebra $x$ and $and$ that do not commute; The Baker-Campbell-Hausdorff formula provides the necessary correction terms.

Transcendence

Function $e z$ is not in $C (z)$ (ie, not the quotient of two polynomials with complex coefficients).

For $n$ distinct complex numbers ${a 1, \dots, a n$ }, the set ${e a 1 z, \dots, e a n z$ } is linearly independent over $C (z)$ .

The function $e z$ is trascendental over $C (z)$

Computing

When computing (an approximation of) the exponential function, if the argument is close to 0, the result will be close to 1, and computing the difference ${displaystyle exp(x)-1}$ can cause a loss of precision.

Following a proposal by William Kahan, it may be useful to have a dedicated routine, often called expm1, to compute $e x - 1$ directly, without going through the computation of $e x$ . For example, if the exponential is calculated using its Taylor series

{displaystyle e^{x}=1+x+{frac {x^{2}}{2}}+{frac {x^{3}}{6}}+cdots +{frac {x^{n}}{n!}}+cdots}

One can use the Taylor series ${displaystyle e^{x}-1:}$

{displaystyle e^{x}-1=x+{frac {x^{2}}{2}}+{frac {x^{3}}{6}}+cdots +{frac {x^{n}}{n!}}+cdots.}

This was first implemented in 1979 in the Hewlett-Packard HP-41C calculator, and was provided by various calculators, computer algebra systems, and programming languages (for example, C99).

A similar approach has been used for the logarithm (see lnp1).

An identity in terms of the hyperbolic tangent,

{displaystyle operatorname {expm1} (x)=exp(x)-1={frac {2tanh(x/2)}{1-tanh(x/2)}},}

provides a high-precision value for small values of x on systems that do not implement $expm1(x)$ .

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Exponential function

Formal definition

Overview

Derivatives and differential equations

Continued Fractions for ex

Complex shot

Calculation of ab where both a and b are complex

General exponential function

Banach matrices and algebras

Lie Algebras

Transcendence

Computing

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