The exponential function is a mathematical function denoted by displaystylef(x)=exp(x) or displaystyleeˣ (where the argument x is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. The exponential function originated from the operation of taking powers of a number (repeated multiplication), but various modern definitions allow it to be rigorously extended to all real arguments displaystyle x, including irrational numbers. Its ubiquitous occurrence in pure and applied mathematics led mathematician Walter Rudin to consider the exponential function to be "the most important function in mathematics". The function displaystylef(x)=bˣ for any positive real number displaystyle b (called the base) is also known as a (general) exponential function, and satisfies the exponentiation identity:displaystylebˣ⁺ʸ=bˣbʸtextforallx,yinmathbbR.This implies displaystylebⁿ=btimes cdots times b (with displaystyle n factors) for positive integers displaystyle n, where displaystyleb=b¹, relating exponential functions to the elementary notion of exponentiation. The natural base displaystylee=exp(1)=2.71828ldots is a ubiquitous mathematical constant called Euler's number. To distinguish it, displaystyleexp(x)=eˣ is called the exponential function or the natural exponential function: it is the unique real-valued function of a real variable whose derivative is itself and whose value at 0 is 1: The relation displaystylebˣ=eˣˡⁿ ᵇ for displaystyleb>0 and real or complex displaystyle x allows general exponential functions to be expressed in terms of the natural exponential. More generally, especially in applied settings, any function displaystylef:mathbbRtomathbbR defined by displaystylef(x)=a,eᵏˣ=a,bᶜˣ,textwithk=clnb,kneq0,a,b>0 is also known as an exponential function: it solves the initial value problem displaystylefʼ=kf,f(0)=a, meaning its rate of change at each point is proportional to the value of the function at that point. This behavior models diverse phenomena in the biological, physical, and social sciences, for example the unconstrained growth of a self-reproducing population, the decay of a radioactive element, the compound interest accruing on a financial fund, or the self-sustaining improvement of computer design. The exponential function can also be defined as a power series, which is readily applied to real, complex, and even matrix arguments. The complex exponential function displaystyleexp:mathbbCtomathbbC takes on all complex values except 0 and is closely related to the trigonometric functions by Euler's formula: Motivated by its more abstract properties and characterizations, the exponential function can be generalized to much larger contexts such as square matrices and Lie groups. Even further, the differential equation definition can be generalized to a Riemannian manifold. The exponential function for real numbers is a bijection from displaystylemathbbR to the interval displaystyle(0,infty). Its inverse function is the natural logarithm, denoted displaystyleln, displaystylelog, or displaystylelogₑ. Some old texts refer to the exponential function as the antilogarithm. The graph of displaystyley=eˣ is upward-sloping, and increases faster as x increases. The graph always lies above the x-axis, but becomes arbitrarily close to it for large negative x; thus, the x-axis is a horizontal asymptote. The equation displaystyletfracddxeˣ=eˣ means that the slope of the tangent to the graph at each point is equal to its height (its y-coordinate) at that point. The exponential function displaystylef(x)=eˣ is sometimes called the natural exponential function to distinguish it from the other exponential functions. The study of any exponential function can easily be reduced to that of the natural exponential function, since per definition, for positive b, displaystylea,bˣmathrelstackreltextdef=a,eˣˡⁿ ᵇAs functions of a real variable, exponential functions are uniquely characterized by the fact that the derivative of such a function is directly proportional to the value of the function. The constant of proportionality of this relationship is the natural logarithm of the base b: displaystylefracddxa,bˣ=fracddxa,eˣˡⁿ⁽ᵇ⁾=a,eˣˡⁿ⁽ᵇ⁾ln(b)=a,bˣln(b).Let displaystylea>0 be a positive coefficient. For displaystyleb>1, the function displaystylea,bˣ is increasing (as depicted for b = e and b = 2), because displaystylelnb>0 makes the derivative always positive, and describes exponential growth. For displaystyle0<b<1, the function is decreasing (as depicted for b = 1/2), and describes exponential decay. For b = 1, the function is constant. Euler's number e = 2.71828... is the unique base for which the constant of proportionality is 1, since displaystyleln(e)=1, so that the function is its own derivative: displaystylefracddxeˣ=eˣln(e)=eˣ. This function, also denoted as displaystyleexp(x), is called the "natural exponential function", or simply "the exponential function", denoted asdisplaystylexmapstoeˣquadtextorquadxmapstoexp(x).The former notation is commonly used for simpler exponents, while the latter is preferred when the exponent is more complicated and harder to read in a small font. Since any exponential function displaystylef(x)=a,bˣ can be written in terms of the natural exponential, it is computationally and conceptually convenient to reduce the study of exponential functions to this particular one. For real numbers displaystylec,d, a function of the form displaystylef(x)=abᶜˣ⁺ᵈ is also an exponential function: displaystyleabᶜˣ⁺ᵈ=left(abᵈright)left(bᶜright)ˣ=left(abᵈright)eˣᶜˡⁿ ᵇ. The exponential function displaystyleexp can be characterized in a variety of equivalent ways. It is commonly defined by the following power series: displaystyleexpx:=sumₖ₌₀ⁱⁿᶠᵗʸfracxᵏk!=1+x+fracx²2!+fracx³3!+fracx⁴4!+cdots Since the radius of convergence of this power series is infinite, this definition is applicable to all complex numbers; see § Complex plane. The term-by-term differentiation of this power series reveals that textstylefracddxexpx=exp x for all x, leading to another common characterization of displaystyle exp x as the unique solution of the ordinary differential equation displaystyleyʼ(x)=y(x) that satisfies the initial condition displaystyley(0)=1. The same differential equation displaystyleyʼ(x)=y(x), displaystyleyʼ(0)=1 can also be solved using Euler's method, which gives another common characterization, the product limit formula: displaystyleexpx=limₙₜₒᵢₙfₜyleft(1+fracxnright)ⁿ. With any of these equivalent definitions, one defines Euler's number textstylee=exp1. It can then be shown that textstyle(exp1)ˣ is equal to the exponential function textstyle exp x, and both can be written as textstyleeˣ. There is also another way to characterize the exponential function for real numbers: it is the unique function displaystylef:mathbbRtomathbbR that satisfies the identity displaystylef(x+y)=f(x)f(y) for all real displaystylex,y, takes the value displaystylef(1)=e, and attains any of the following regularity conditions: displaystyle f is continuous anywhere; displaystyle f is increasing over any interval; displaystyle f is bounded over any interval. In larger domains, i.e. the complex numbers, the above conditions do not suffice to uniquely characterize displaystylef(x)equiveˣ for all displaystyle x. One may use stronger conditions, such as the complex derivative displaystylefʼ(0)=1. The exponential function arises whenever a quantity grows or decays at a rate proportional to its current value. One such situation is continuously compounded interest, and in fact it was this observation that led Jacob Bernoulli in 1683 to the number displaystylelimₙₜₒᵢₙfₜyleft(1+frac1nright)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 compounded monthly, 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 interest is compounded daily, this becomes (1 + x/365)³⁶⁵. Letting the number of time intervals per year grow without bound leads to the limit definition of the exponential function, displaystyleexpx=limₙₜₒᵢₙfₜyleft(1+fracxnright)ⁿ first given by Leonhard Euler. This is one of a number of characterizations of the exponential function; others involve series or differential equations. From any of these definitions it can be shown that e is the reciprocal of e. For example, from the differential equation definition, e e = 1 when x = 0 and its derivative using the product rule is e e − e e = 0 for all x, so e e = 1 for all x. From any of these definitions it can be shown that the exponential function obeys the basic exponentiation identity. For example, from the power series definition, expanded by the Binomial theorem, displaystyleexp(x+y)=sumₙ₌₀ⁱⁿᶠᵗʸfrac(x+y)ⁿn!=sumₙ₌₀ⁱⁿᶠᵗʸsumₖ₌₀ⁿfracn!k!(n-k)!fracxᵏyⁿ⁻ᵏn!=sumₖ₌₀ⁱⁿᶠᵗʸsumₑₗₗ₌₀ⁱⁿᶠᵗʸfracxᵏyᵉˡˡk!ell!=expxcdotexpy,. This justifies the exponential notation e for exp x. The derivative of the exponential function is the exponential function itself. More generally, a function with a rate of change proportional to the function itself is expressible in terms of the exponential function. This derivative property leads to exponential growth or exponential decay. The exponential function extends to an entire function on the complex plane. Euler's formula relates its values at purely imaginary arguments to trigonometric functions. The exponential function also has analogues for which the argument is a matrix, or even an element of a Banach algebra or a Lie algebra. The importance of the exponential function in mathematics and the sciences stems mainly from its property as the unique function which is equal to its derivative and is equal to 1 when x = 0. That is, displaystylefracddxeˣ=eˣquadtextandquade⁰=1. Functions of the form ae for constant a are the only functions that are equal to their derivative. 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. exp is a fixed point of derivative as a functional. If a variable's growth or decay rate is proportional to its size—as is the case in unlimited population growth, continuously compounded interest, or radioactive decay—then the variable can be written as a constant times an exponential function of time. More generally, for any real constant k, a function f: R → R satisfies displaystylefʼ=kf if and only if displaystylef(x)=aeᵏˣ for some constant a. The constant k is called the decay constant, disintegration constant, rate constant, or transformation constant. Furthermore, for any differentiable function f, we find, by the chain rule: displaystylefracddxeᶠ⁽ˣ⁾=fʼ(x),eᶠ⁽ˣ⁾. A continued fraction for e can be obtained via an identity of Euler: displaystyleeˣ=1+cfracx1-cfracxx+2-cfrac2xx+3-cfrac3xx+4-ddots The following generalized continued fraction for e converges more quickly: displaystyleeᶻ=1+cfrac2z2-z+cfracz²6+cfracz²10+cfracz²14+ddots or, by applying the substitution z = x/y: displaystyleeᶠʳᵃᶜˣʸ=1+cfrac2x2y-x+cfracx²6y+cfracx²10y+cfracx²14y+ddots with a special case for z = 2: displaystylee²=1+cfrac40+cfrac2²6+cfrac2²10+cfrac2²14+ddots,=7+cfrac25+cfrac17+cfrac19+cfrac111+ddots, This formula also converges, though more slowly, for z > 2. For example: displaystylee³=1+cfrac6-1+cfrac3²6+cfrac3²10+cfrac3²14+ddots,=13+cfrac547+cfrac914+cfrac918+cfrac922+ddots, As in the real case, the exponential function can be defined on the complex plane in several equivalent forms. The most common definition of the complex exponential function parallels the power series definition for real arguments, where the real variable is replaced by a complex one: displaystyleexpz:=sumₖ₌₀ⁱⁿᶠᵗʸfraczᵏk! Alternatively, the complex exponential function may be defined by modelling the limit definition for real arguments, but with the real variable replaced by a complex one: displaystyleexpz:=limₙₜₒᵢₙfₜyleft(1+fracznright)ⁿ For the power series definition, term-wise multiplication of two copies of this power series in the Cauchy sense, permitted by Mertens' theorem, shows that the defining multiplicative property of exponential functions continues to hold for all complex arguments: displaystyleexp(w+z)=expwexpztextforallw,zinmathbbC The definition of the complex exponential function in turn leads to the appropriate definitions extending the trigonometric functions to complex arguments. In particular, when z = it (t real), the series definition yields the expansion displaystyleexp(it)=left(1-fract²2!+fract⁴4!-fract⁶6!+cdots right)+ileft(t-fract³3!+fract⁵5!-fract⁷7!+cdots right). In this expansion, the rearrangement of the terms into real and imaginary parts is justified by the absolute convergence of the series. The real and imaginary parts of the above expression in fact correspond to the series expansions of cos t and sin t, respectively. This correspondence provides motivation for defining cosine and sine for all complex arguments in terms of displaystyleexp(pm iz) and the equivalent power series: displaystylebeginaligned&cosz:=fracexp(iz)+exp(-iz)2=sumₖ₌₀ⁱⁿᶠᵗʸ(-1)ᵏfracz²ᵏ(2k)!,[5pt]textandquad&sinz:=fracexp(iz)-exp(-iz)2i=sumₖ₌₀ⁱⁿᶠᵗʸ(-1)ᵏfracz²ᵏ⁺¹(2k+1)!endaligned for all textstylezinmathbbC. The functions exp, cos, and sin so defined have infinite radii of convergence by the ratio test and are therefore entire functions (that is, holomorphic on displaystylemathbbC). The range of the exponential function is displaystylemathbbCsetminus0, while the ranges of the complex sine and cosine functions are both displaystylemathbbC in its entirety, in accord with Picard's theorem, which asserts that the range of a nonconstant entire function is either all of displaystylemathbbC, or displaystylemathbbC excluding one lacunary value. These definitions for the exponential and trigonometric functions lead trivially to Euler's formula: displaystyleexp(iz)=cosz+isinztextforallzinmathbbC. We could alternatively define the complex exponential function based on this relationship. If z = x + iy, where x and y are both real, then we could define its exponential as displaystyleexpz=exp(x+iy):=(exp x)(cosy+isin 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. For displaystyletinmathbbR, the relationship displaystyleoverlineexp(it)=exp(-it) holds, so that displaystyleleft|exp(it)right|=1 for real displaystyle t and displaystyletmapstoexp(it) maps the real line (mod 2π) to the unit circle in the complex plane. Moreover, going from displaystylet=0 to displaystylet=t₀, the curve defined by displaystylegamma(t)=exp(it) traces a segment of the unit circle of length displaystyleint₀ᵗ₀|gammaʼ(t)|,dt=int₀ᵗ₀|iexp(it)|,dt=t₀, starting from z = 1 in the complex plane and going counterclockwise. Based on these observations and the fact that the measure of an angle in radians is the arc length on the unit circle subtended by the angle, it is easy to see that, restricted to real arguments, the sine and cosine functions as defined above coincide with the sine and cosine functions as introduced in elementary mathematics via geometric notions. The complex exponential function is periodic with period 2πi and displaystyleexp(z+2pi ik)=exp z holds for all displaystylezinmathbbC,kinmathbbZ. When its domain is extended from the real line to the complex plane, the exponential function retains the following properties: displaystylebeginaligned&eᶻ⁺ʷ=eᶻeʷ,[5pt]&e⁰=1,[5pt]&eᶻneq0[5pt]&fracddzeᶻ=eᶻ[5pt]&left(eᶻright)ⁿ=eⁿᶻ,ninmathbbZendaligned for all textstylew,zinmathbbC. Extending the natural logarithm to complex arguments yields the complex logarithm log z, which is a multivalued function. We can then define a more general exponentiation: displaystylezʷ=eʷˡᵒᵍ ᶻ for all complex numbers z and w. This is also a multivalued function, even when z is real. This distinction is problematic, as the multivalued functions log z and z 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: See failure of power and logarithm identities for more about problems with combining powers. 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 exist: when the original line is parallel to the real axis, the resulting spiral never closes in on itself; when the original line is parallel to the imaginary axis, the resulting spiral is a circle of some radius. 3D plots of real part, imaginary part, and modulus of the exponential function z = Re(e) z = Im(e) z = |e| Considering the complex exponential function as a function involving four real variables: displaystylev+iw=exp(x+iy) the graph of the exponential function is a two-dimensional surface curving through four dimensions. Starting with a color-coded portion of the displaystyle xy domain, the following are depictions of the graph as variously projected into two or three dimensions. Graphs of the complex exponential function Checker board key: displaystylex>0:;textgreen displaystylex<0:;textreddisplaystyley>0:;textyellowdisplaystyley<0:;textblue Projection onto the range complex plane (V/W). Compare to the next, perspective picture. Projection into the displaystyle x, displaystyle v, and displaystyle w dimensions, producing a flared horn or funnel shape (envisioned as 2-D perspective image) Projection into the displaystyle y, displaystyle v, and displaystyle w dimensions, producing a spiral shape (displaystyle y range extended to ±2π, again as 2-D perspective image) The second image shows how the domain complex plane is mapped into the range complex plane: zero is mapped to 1 the real displaystyle x axis is mapped to the positive real displaystyle v axis the imaginary displaystyle y axis is wrapped around the unit circle at a constant angular rate values with negative real parts are mapped inside the unit circle values with positive real parts are mapped outside of the unit circle values with a constant real part are mapped to circles centered at zero values with a constant imaginary part are mapped to rays extending from zero The third and fourth images show how the graph in the second image extends into one of the other two dimensions not shown in the second image. The third image shows the graph extended along the real displaystyle x axis. It shows the graph is a surface of revolution about the displaystyle x axis of the graph of the real exponential function, producing a horn or funnel shape. The fourth image shows the graph extended along the imaginary displaystyle y axis. It shows that the graph's surface for positive and negative displaystyle y values doesn't really meet along the negative real displaystyle v axis, but instead forms a spiral surface about the displaystyle y axis. Because its displaystyle y values have been extended to ±2π, this image also better depicts the 2π periodicity in the imaginary displaystyle y value. Complex exponentiation a can be defined by converting a to polar coordinates and using the identity (e) = a: displaystyleaᵇ=left(reᵗʰᵉᵗᵃ ⁱright)ᵇ=left(e⁽ˡⁿ ʳ⁾⁺ᵗʰᵉᵗᵃ ⁱright)ᵇ=eˡᵉᶠᵗ⁽⁽ˡⁿ ʳ⁾⁺ᵗʰᵉᵗᵃ ⁱʳⁱᵍʰᵗ⁾ᵇ However, when b is not an integer, this function is multivalued, because θ is not unique. The power series definition of the exponential function makes sense for square matrices and more generally in any unital Banach algebra B. In this setting, e⁰ = 1, and e is invertible with inverse e for any x in B. If xy = yx, then e = ee, but this identity can fail for noncommuting x and y. Some alternative definitions lead to the same function. For instance, e can be defined as displaystylelimₙₜₒᵢₙfₜyleft(1+fracxnright)ⁿ. Or e can be defined as fₓ, where fₓ: R → B is the solution to the differential equation dfₓ/dt(t) = x fₓ(t), with initial condition fₓ = 1; it follows that fₓ(t) = e for every t in R. Given a Lie group G and its associated Lie algebra displaystylemathfrakg, the exponential map is a map displaystylemathfrakg ↦ G satisfying similar properties. In fact, since R is the Lie algebra of the Lie group of all positive real numbers under multiplication, the ordinary exponential function for real arguments is a special case of the Lie algebra situation. Similarly, since the Lie group GL(n,R) of invertible n × n matrices has as Lie algebra M(n,R), the space of all n × n matrices, the exponential function for square matrices is a special case of the Lie algebra exponential map. The identity displaystyleexp(x+y)=exp(x)exp(y) can fail for Lie algebra elements x and y that do not commute; the Baker–Campbell–Hausdorff formula supplies the necessary correction terms. The function e is not in the rational function ring displaystylemathbbC(z): it is not the quotient of two polynomials with complex coefficients. If a₁, ..., aₙ are distinct complex numbers, then e, ..., e are linearly independent over displaystylemathbbC(z), and hence e is transcendental over displaystylemathbbC(z). When computing (an approximation of) the exponential function near the argument 0, the result will be close to 1, and computing the value of the difference displaystyleeˣ-1 with floating-point arithmetic may lead to the loss of (possibly all) significant figures, producing a large calculation error, possibly even a meaningless result. Following a proposal by William Kahan, it may thus be useful to have a dedicated routine, often called expm1, for computing eˣ − 1 directly, bypassing computation of e. For example, if the exponential is computed by using its Taylor series displaystyleeˣ=1+x+fracx²2+fracx³6+cdots+fracxⁿn!+cdots, one may use the Taylor series of displaystyleeˣ-1: displaystyleeˣ-1=x+fracx²2+fracx³6+cdots+fracxⁿn!+cdots. This was first implemented in 1979 in the Hewlett-Packard HP-41C calculator, and provided by several calculators, operating systems, computer algebra systems, and programming languages. In addition to base e, the IEEE 754-2008 standard defines similar exponential functions near 0 for base 2 and 10: displaystyle2ˣ-1 and displaystyle10ˣ-1. A similar approach has been used for the logarithm.
