Exponential function

The exponential function is a function in mathematics. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The application of this function to a value x is written as exp(x). Equivalently, this can be written in the form e^x, where e is a mathematical constant, the base of the natural logarithm, which equals approximately 2. The Mathematical constant e is the unique Real number such that the function e x has the same value as the slope of the tangent line 718281828, and is also known as Euler's number.

The exponential function is nearly flat (climbing slowly) for negative values of x, climbs quickly for positive values of x, and equals 1 when x is equal to 0. Its y value always equals the slope at that point. Slope is used to describe the steepness incline gradient or grade of a straight line.

As a function of the real variable x, the graph of y=e^x is always positive (above the x axis) and increasing (viewed left-to-right). In Mathematics, the real numbers may be described informally in several different ways In mathematics the graph of a function f is the collection of all Ordered pairs ( x, f ( x) It never touches the x axis, although it gets arbitrarily close to it (thus, the x axis is a horizontal asymptote to the graph). An asymptote of a real-valued function y=f(x is a curve which describes the behavior of f as either x or y goes to infinity Its inverse function, the natural logarithm, ln(x), is defined for all positive x. In Mathematics, if &fnof is a function from A to B then an inverse function for &fnof is a function in the opposite direction from B The natural logarithm, formerly known as the Hyperbolic logarithm is the Logarithm to the base e, where e is an irrational The exponential function is occasionally referred to as the anti-logarithm. In Mathematics, the logarithm of a number to a given base is the power or Exponent to which the base must be raised in order to produce However, this terminology seems to have fallen into disuse in recent times.

Sometimes, especially in the sciences, the term exponential function is more generally used for functions of the form ka^x, where a, called the base, is any positive real number not equal to one. Science (from the Latin scientia, meaning " Knowledge " or "knowing" is the effort to discover, and increase human understanding This article will focus initially on the exponential function with base e, Euler's number.

In general, the variable x can be any real or complex number, or even an entirely different kind of mathematical object; see the formal definition below. A variable (ˈvɛərɪəbl is an Attribute of a physical or an abstract System which may change its Value while it is under Observation. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted

1 Properties
2 Derivatives and differential equations
3 Formal definition
4 Numerical value
5 Computing exp(x) for real x
6 Continued fractions for e^x
7 On the complex plane
8 Computation of exp(z) for a complex z
9 Computation of where both a and b are complex
10 Matrices and Banach algebras
11 On Lie algebras
12 Double exponential function
13 Similar properties of e and the function e^z
14 Periodicity
15 References
16 See also
17 External links

Properties

Most simply, exponential functions multiply at a constant rate. For example the population of a bacterial culture which doubles every 20 minutes can (approximatively, as this is not really a continuous problem) be expressed as an exponential, as can the value of a car which decreases by 10% per year.

Using the natural logarithm, one can define more general exponential functions. The function

$\,\!\, a^x=(e^{\ln a})^x=e^{x \ln a}$

defined for all a > 0, and all real numbers x, is called the exponential function with base a. Note that this definition of $\, a^x$ rests on the previously established existence of the function $\, e^x$ , defined for all real numbers. (Here, we neither formally nor conceptually clarify whether such a function exists or what non-natural exponents are supposed to mean. )

Note that the equation above holds for a = e, since

$\,\!\, e^{x \ln e}=e^{x \cdot 1}=e^x.$

Exponential functions "translate between addition and multiplication" as is expressed in the first three and the fifth of the following exponential laws:

$\,\!\, a^0 = 1$

$\,\!\, a^1 = a$

$\,\!\, a^{x + y} = a^x a^y$

$\,\!\, a^{x y} = \left( a^x \right)^y$

$\,\!\, {1 \over a^x} = \left({1 \over a}\right)^x = a^{-x}$

$\,\!\, a^x b^x = (a b)^x$

These are valid for all positive real numbers a and b and all real numbers x and y. Expressions involving fractions and roots can often be simplified using exponential notation:

$\,{1 \over a} = a^{-1}$

and, for any a > 0, real number b, and integer n > 1:

$\,\sqrt[n]{a^b} = \left(\sqrt[n]{a}\right)^b = a^{b/n}.$

Derivatives and differential equations

The importance of exponential functions in mathematics and the sciences stems mainly from properties of their derivatives. In Mathematics, a fraction (from the Latin fractus, broken is a concept of a proportional relation between an object part and the object In Mathematics, an n th root of a Number a is a number b such that bn = a. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In particular,

$\,{d \over dx} e^x = e^x.$

That is, e^x is its own derivative. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change Functions of the form $\,Ke^x$ for constant K are the only functions with that property. (This follows from the Picard-Lindelöf theorem, with $\,y(t) = e^t, y(0)=K$ and $\,f(t,y(t)) = y(t)$ . ) Other ways of saying the same thing include:

The slope of the graph at any point is the height of the function at that point.
The rate of increase of the function at x is equal to the value of the function at x.
The function solves the differential equation $\,y'=y$ . A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the
exp is a fixed point of derivative as a functional

In fact, many differential equations give rise to exponential functions, including the Schrödinger equation and the Laplace's equation as well as the equations for simple harmonic motion. In Mathematics, a functional is traditionally a map from a Vector space to the field underlying the vector space which is usually the Real In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system In Mathematics, Laplace's equation is a Partial differential equation named after Pierre-Simon Laplace who first studied its properties Simple harmonic motion is the motion of a simple harmonic oscillator, a motion that is neither driven nor damped.

For exponential functions with other bases:

$\,{d \over dx} a^x = (\ln a) a^x.$

Thus, any exponential function is a constant multiple of its own derivative.

If a variable's growth or decay rate is proportional to its size — as is the case in unlimited population growth (see Malthusian catastrophe), continuously compounded interest, or radioactive decay — then the variable can be written as a constant times an exponential function of time. This article is about proportionality the mathematical relation A Malthusian catastrophe (or Malthusian check, crisis, dilemma, disaster, trap, controls, or limit) is a return Interest is a fee paid on borrowed capital Assets lent include Money, Shares, Consumer goods through Hire purchase, major assets Radioactive decay is the process in which an unstable Atomic nucleus loses energy by emitting ionizing particles and Radiation.

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

$\,{d \over dx} e^{f(x)} = f'(x)e^{f(x)}.$

Formal definition

The exponential function (in blue), and the sum of the first n+1 terms of the power series on the left (in red). In Calculus, the chain rule is a Formula for the Derivative of the composite of two functions.

The exponential function e^x can be defined in a variety of equivalent ways, as an infinite series. In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with In particular it may be defined by a power series:

$e^x = \sum_{n = 0}^{\infty} {x^n \over n!} = 1 + x + {x^2 \over 2!} + {x^3 \over 3!} + {x^4 \over 4!} + \cdots$ . In Mathematics, a power series (in one variable is an Infinite series of the form f(x = \sum_{n=0}^\infty a_n \left( x-c \right^n = a_0 +

Note that this definition has the form of a Taylor series. In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives Using an alternate definition for the exponential function should lead to the same result when expanded as a Taylor series. In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives

A less common definition defines $e x$ as the solution $y$ to the equation

$x = \int_{1}^y {dt \over t}.$

It can also be considered to be the following limit:

$e^x = \lim_{n \rightarrow \infty} \left(1 + \frac{x}{n}\right)^{n}$

Numerical value

To obtain the numerical value of the exponential function, the infinite series can be rewritten as :

$\,e^x = {1 \over 0!} + x \, \left( {1 \over 1!} + x \, \left( {1 \over 2!} + x \, \left( {1 \over 3!} + \cdots \right)\right)\right)$

$\,= 1 + {x \over 1} \left(1 + {x \over 2} \left(1 + {x \over 3} \left(1 + \cdots \right)\right)\right)$

This expression will converge quickly if we can ensure that x is less than one.

To ensure this, we can use the following identity.

$\,e^x\,$	$\,=e^{z+f}\,$
	$\,= e^z \times \left[{1 \over 0!} + f \, \left( {1 \over 1!} + f \, \left( {1 \over 2!} + f \, \left( {1 \over 3!} + \cdots \right)\right)\right)\right]$

Where $\,z$ is the integer part of $\,x$
Where $\,f$ is the fractional part of $\,x$
Hence, $\,f$ is always less than 1 and $\,f$ and $\,z$ add up to $\,x$ .

The value of the constant e^z can be calculated beforehand by multiplying e with itself z times.

Computing exp(x) for real x

An even better algorithm can be found as follows.

First, notice that the answer y = e^x is usually a floating point number represented by a mantissa m and an exponent n so y = m 2ⁿ for some integer n and suitably small m. Thus, we get:

$\,y = m\,2^n = e^x.$

Taking log on both sides of the last two gives us:

$\,\ln(y) = \ln(m) + n\ln(2) = x.$

Thus, we get n as the result of dividing x by log(2) and finding the greatest integer that is not greater than this - that is, the floor function:

$\,n = \left\lfloor\frac{x}{\ln(2)}\right\rfloor.$

Having found n we can then find the fractional part u like this:

$\,u = x - n\ln(2).$

The number u is small and in the range 0 ≤ u < ln(2) and so we can use the previously mentioned series to compute m:

$\,m = e^u = 1 + u(1 + u(\frac{1}{2!} + u(\frac{1}{3!} + u(....)))).$

Having found m and n we can then produce y by simply combining those two into a floating point number:

$\,y = e^x = m\,2^n.$

Continued fractions for e^x

Via Euler's identity:

$\,\ e^x=1+x+\frac{x^2}{2!}+\cdots=1+\cfrac{x}{1-\cfrac{x}{x+2-\cfrac{2x}{x+3-\cfrac{3x}{x+4-\cfrac{4x}{x+5-\cfrac{5x}{\ddots}}}}}}$

More advanced techniques are necessary to construct the following:

$\,\ e^{2m/n}=1+\cfrac{2m}{(n-m)+\cfrac{m^2}{3n+\cfrac{m^2}{5n+\cfrac{m^2}{7n+\cfrac{m^2}{9n+\cfrac{m^2}{\ddots}}}}}}\,$

Setting m = x and n = 2 yields

$\,\ e^x=1+\cfrac{2x}{(2-x)+\cfrac{x^2}{6+\cfrac{x^2}{10+\cfrac{x^2}{14+\cfrac{x^2}{18+\cfrac{x^2}{\ddots}}}}}}\,$

On the complex plane

Exponential function on the complex plane. In Mathematics and Computer science, the floor and ceiling functions map Real numbers to nearby Integers The The transition from dark to light colors shows that the magnitude of the exponential function is increasing to the right. The 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 on the complex plane in several equivalent forms. In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted Some of these definitions mirror the formulas for the real-valued exponential function. Specifically, one can still use the power series definition, where the real value is replaced by a complex one:

$\,\!\, e^z = \sum_{n = 0}^\infty\frac{z^n}{n!}$

Using this definition, it is easy to show why ${d \over dz} e^z = e^z$ holds in the complex plane.

Another definition extends the real exponential function. In Mathematics, the real numbers may be described informally in several different ways First, we state the desired property $e x + i y = e x e i y$ . For $e x$ we use the real exponential function. In Mathematics, the real numbers may be described informally in several different ways We then proceed by defining only: $e i y = c o s (y) + i s i n (y)$ . Thus we use the real definition rather than ignore it. In Mathematics, the real numbers may be described informally in several different ways ^[1]

When considered as a function defined on the complex plane, the exponential function retains the important properties

$\,\!\, e^{z + w} = e^z e^w$

$\,\!\, e^0 = 1$

$\,\!\, e^z \ne 0$

$\,\!\, {d \over dz} e^z = e^z$

for all z and w. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted

It is a holomorphic function which is periodic with imaginary period $\,2 \pi i$ and can be written as

$\,\!\, e^{a + bi} = e^a (\cos b + i \sin b)$

where a and b are real values. Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane Geometric interpretation Geometrically imaginary numbers are found on the vertical axis of the complex number plane This formula connects the exponential function with the trigonometric functions and to the hyperbolic functions. In Mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular functions Thus we see that all elementary functions except for the polynomials spring from the exponential function in one way or another. This article discusses the concept of elementary functions in differential algebra In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations

See also Euler's formula. This article is about Euler's formula in Complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic

Extending the natural logarithm to complex arguments yields a multi-valued function, ln(z). In Mathematics, a multivalued function (shortly multifunction, other names set-valued function, set-valued map, multi-valued map We can then define a more general exponentiation:

$\,\!\, z^w = e^{w \ln z}$

for all complex numbers z and w. This is also a multi-valued function. The above stated exponential laws remain true if interpreted properly as statements about multi-valued functions.

The exponential function maps any line in the complex plane to a logarithmic spiral in the complex plane with the center at the origin. Definition In Polar coordinates ( r, θ the curve can be written as r = ae^{b\theta}\ or \theta In Mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference Two special cases might be noted: when the original line is parallel to the real axis, the resulting sprial never closes in on itself; when the original line is parallel to the imaginary axis, the resulting spiral is a circle of some radius.

Plots of the exponential function on the complex plane
z = Re(e^x+iy)	z = Im(e^x+iy)	z = \|e^x+iy\|

Computation of exp(z) for a complex z

This is fairly straightforward given the formula

$\,e^{x + yi} = e^xe^{yi} = e^x(\cos(y) + i \sin(y)) = e^x\cos(y) + ie^x\sin(y).$

Note that the argument y to the trigonometric functions is real.

Computation of $\,a^b$ where both a and b are complex

This is also straightforward given the formulae:

if a = x + yi and b = u + vi we can first convert a to polar co-ordinates by finding a $\,\theta$ and an $\,r$ such that:

$\,re^{{\theta}i} = r\cos\theta + i r\sin\theta = a = x + yi$

$\, x = r\cos\theta$ and $\,y = r\sin\theta.$

Thus, $\,x^2 + y^2 = r^2$ or $\,r = \sqrt{x^2 + y^2}$ and $\,\tan\theta = \frac{y}{x}$ or $\,\theta = \operatorname{atan2}(y, x).$

Now, we have that:

$\,a = re^{{\theta}i} = e^{\ln(r) + {\theta}i}$

so:

$\,a^b = (e^{\ln(r) + {\theta}i})^{u + vi} = e^{(\ln(r) + {\theta}i)(u + vi)}$

The exponent is thus a simple multiplication of two complex values yielding a complex result which can then be brought back to regular cartesian format by the formula:

$\,e^{p + qi} = e^p(\cos(q) + i\sin(q)) = e^p\cos(q) + ie^p\sin(q)$

where p is the real part of the multiplication:

$\,p = u\ln(r) - v\theta$

and q is the imaginary part of the multiplication:

$\,q = v\ln(r) + u\theta.$

Note that all of $\,x, y, u, v, r,$ $\,\theta$ , $\,p$ and $\,q$ are all real values in these computations.

Also note that since we compute and use $\,\ln(r)$ rather than r itself you don't have to compute the square root. Instead simply compute $\,\ln(r) = \frac12\ln(x^2 + y^2)$ . Watch out for potential overflow though and possibly scale down the x and y prior to computing $\,x^2 + y^2$ by a suitable power of 2 if $\,x$ and $\,y$ are so large that you would overflow. If you instead run the risk of underflow, scale up by a suitable power of 2 prior to computing the sum of the squares. In either case you then get the scaled version of $\,x$ - we can call it $\,x'$ and the scaled version of $\,y$ - call it $\,y'$ and so you get:

$\,x = x'2^s$ and $\,y = y'2^s$

where $\,2^s$ is the scaling factor.

Then you get $\,\ln(r) = \frac12(\ln(x'^2 + y'^2) + s)$ where $\,x'$ and $\,y'$ are scaled so that the sum of the squares will not overflow or underflow. If $\,x$ is very large while $\,y$ is very small so that you cannot find such a scaling factor you will overflow anyway and so the sum is essentially equal to $\,x^2$ since y is ignored and thus you get $\,r = |x|$ in this case and $\,\ln(r) = \log(|x|)$ . The same happens in the case when $\,x$ is very small and $\,y$ is very large. If both are very large or both are very small you can find a scaling factor as mentioned earlier.

Note that this function is, in general, multivalued for complex arguments. In Mathematics, a multivalued function (shortly multifunction, other names set-valued function, set-valued map, multi-valued map This is because rotation of a single point through any angle plus 360 degrees, or $2π$ radians, is the same as rotation through the angle itself. So $θ$ above is not unique: $θ k = θ + 2π k$ for any integer $k$ would do as well. The convention though is that when $a b$ is taken as a single value it must be that for $k = 0$ , ie. we use the smallest possible (in magnitude) value of theta, which has a magnitude of, at most, $π$ .

Matrices and Banach algebras

The definition of the exponential function given above can be used verbatim for every Banach algebra, and in particular for square matrices (in which case the function is called the matrix exponential). In Mathematics, especially Functional analysis, a Banach algebra, named after Stefan Banach, is an Associative algebra A over the In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Mathematics, the matrix exponential is a Matrix function on square matrices analogous to the ordinary Exponential function. In this case we have

$\,\ e^{x + y} = e^x e^y \mbox{ if } xy = yx$

$\,\ e^0 = 1$

$\,\ e^x$ is invertible with inverse $\,\ e^{-x}$

the derivative of $\,\ e^x$ at the point $\,\ x$ is that linear map which sends $\,\ u$ to $\,\ ue^x$ .

In the context of non-commutative Banach algebras, such as algebras of matrices or operators on Banach or Hilbert spaces, the exponential function is often considered as a function of a real argument:

$\,\ f(t) = e^{t A}$

where A is a fixed element of the algebra and t is any real number. In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis This article assumes some familiarity with Analytic geometry and the concept of a limit. This function has the important properties

$\,\ f(s + t) = f(s) f(t)$

$\,\ f(0) = 1$

$\,\ f'(t) = A f(t)$

On Lie algebras

The exponential map sending a Lie algebra to the Lie group that gave rise to it shares the above properties, which explains the terminology. In differential geometry the exponential map is a generalization of the ordinary Exponential function of mathematical analysis to all differentiable manifolds with an Affine In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In fact, since R is the Lie algebra of the Lie group of all positive real numbers with multiplication, the ordinary exponential function for real arguments is a special case of the Lie algebra situation. Similarly, since the Lie algebra M (n, R) of all square real matrices belongs to the Lie group of all invertible square matrices, the exponential function for square matrices is a special case of the Lie algebra exponential map. In general, when the argument of the exponential function is noncommutative, the formula is given explicitly by the Baker-Campbell-Hausdorff formula. In Mathematics, the Baker-Campbell-Hausdorff formula is the solution to Z = \log(e^X e^Y\ for non- commuting X

Double exponential function

Main article: double exponential function

The term double exponential function can have two meanings:

a function with two exponential terms, with different exponents
a function $\,f(x) = a^{a^x}$ ; this grows even faster than an exponential function; for example, if a = 10: f(−1) = 1. A double exponential function is a Constant raised to the power of an Exponential function. 26, f(0) = 10, f(1) = 10¹⁰, f(2) = 10¹⁰⁰ = googol, . A googol is the Large number 10100 that is the digit 1 followed by one hundred zeros (in Decimal representation . . , f(100) = googolplex.

Factorials grow faster than exponential functions, but slower than double-exponential functions. Fermat numbers, generated by $\,F(m) = 2^{2^m} + 1$ and double Mersenne numbers generated by $\,MM(p) = 2^{(2^p-1)}-1$ are examples of double exponential functions. In Mathematics, a Fermat number, named after Pierre de Fermat who first studied them is a positive integer of the form F_{n} = 2^{2^{ In Mathematics, a double Mersenne number is a Mersenne number of the form M_{M_p} = 2^{2^p-1}-1 where p is a Mersenne

Similar properties of $e$ and the function $e z$

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

For n distinct complex numbers ${a 1,. . . a n}$ , $\{e^{a_1 z},... e^{a_n z}\}$ is linearly independent over C(z).

The function $e z$ is transcendental over C(z).

Periodicity

For all integers n and complex x:

$e^{x} = e^{x \, \pm \, 2i\pi n}$

Proof:

$\begin{align}e^{x} &= e^{x}1 \\ &= e^{x}1^{\pm n} \\ &= e^{x}(e^{2i\pi})^{\pm n} \\ &= e^{x}e^{\pm 2i\pi n} \\ &= e^{x \, \pm \, 2i\pi n}\end{align}$

For all positive integers n and complex a & x:

$a^{x} = e^{\ln a^{x}} = e^{x \ln a} = e^{x \ln a \, \pm \, 2i\pi n}$

References

^ Ahlfors, Lars V. (1953). Complex analysis. McGraw-Hill Book Company, Inc. .

External links

Complex exponential function on PlanetMath
Derivative of exponential function on PlanetMath
Eric W. Weisstein, Exponential Function at MathWorld. The Mathematical constant e is the unique Real number such that the function e x has the same value as the slope of the tangent line In Mathematics, the Exponential function can be characterized in many ways In Mathematics, tetration (also known as hyper -4 Exponential growth (including Exponential decay) occurs when the growth rate of a mathematical function is proportional to the function's current value The following is a list of Integrals ( Antiderivative functions of Exponential functions For a complete list of Integral functions please see the List of integrals This is a list of exponential topics, by Wikipedia page See also List of logarithm topics. PlanetMath is a free, collaborative online Mathematics Encyclopedia. PlanetMath is a free, collaborative online Mathematics Encyclopedia. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
Complex Exponential Function Module by John H. Mathews
Taylor Series Expansions of Exponential Functions at efunda.com
Complex exponential interactive graphic

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