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Exponential function

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Exponential function

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This article is about functions of the form f(x) = abx. For functions of the form f(x,y) = xy, see Exponentiation. For functions of the form f(x) = xr, see Power function.

The natural exponential function y = ex
Exponential functions with bases 2 and 1/2

In mathematics, an exponential function is a function of the form

f(x)=abx,{\displaystyle f(x)=ab^{x},}

where b is a positive real number, and the argument x occurs as an exponent. For real numbers c and d, a function of the form f(x)=abcx+d{\displaystyle f(x)=ab^{cx+d}} is also an exponential function, since it can be rewritten as

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

As functions of a real variable, exponential functions are uniquely characterized by the fact that the growth rate of such a function (that is, its derivative) is directly proportional to the value of the function. The constant of proportionality of this relationship is the natural logarithm of the base b:

ddxbx=bxloge⁡b.{\displaystyle {\frac {d}{dx}}b^{x}=b^{x}\log _{e}b.}

For b > 1, the function bx{\displaystyle b^{x}} is increasing (as depicted for b = e and b = 2), because 0}”>loge⁡b>0{\displaystyle \log _{e}b>0}0}”> makes the derivative always positive; while for b < 1, the function is decreasing (as depicted for b = .mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px;white-space:nowrap}1/2); and for b = 1 the function is constant.

The constant e = 2.71828… is the unique base for which the constant of proportionality is 1, so that the function is its own derivative:

ddxex=exloge⁡e=ex.{\displaystyle {\frac {d}{dx}}e^{x}=e^{x}\log _{e}e=e^{x}.}

This function, also denoted as exp x, is called the “natural exponential function”,[1][2][3] or simply “the exponential function”. Since any exponential function can be written in terms of the natural exponential as bx=exloge⁡b{\displaystyle b^{x}=e^{x\log _{e}b}}, it is computationally and conceptually convenient to reduce the study of exponential functions to this particular one. The natural exponential is hence denoted by

x↦ex or x↦exp⁡x.{\displaystyle x\mapsto e^{x}{\text{ or }}x\mapsto \exp x.}

The former notation is commonly used for simpler exponents, while the latter is preferred when the exponent is a complicated expression. The graph of y=ex{\displaystyle y=e^{x}} is upward-sloping, and increases faster as x increases.[4] 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 ddxex=ex{\displaystyle {\tfrac {d}{dx}}e^{x}=e^{x}} means that the slope of the tangent to the graph at each point is equal to its y-coordinate at that point. Its inverse function is the natural logarithm, denoted log,{\displaystyle \log ,}[nb 1]ln,{\displaystyle \ln ,}[nb 2] or loge;{\displaystyle \log _{e};} because of this, some old texts[5] refer to the exponential function as the antilogarithm.

The exponential function satisfies the fundamental multiplicative identity (which can be extended to complex-valued exponents as well):

ex+y=exey for all x,y∈R.{\displaystyle e^{x+y}=e^{x}e^{y}{\text{ for all }}x,y\in \mathbb {R} .}

It can be shown that every continuous, nonzero solution of the functional equation f(x+y)=f(x)f(y){\displaystyle f(x+y)=f(x)f(y)} is an exponential function, f:R→R, x↦bx,{\displaystyle f:\mathbb {R} \to \mathbb {R} ,\ x\mapsto b^{x},} with 0.}”>b>0.{\displaystyle b>0.}0.}”> The multiplicative identity, along with the definition e=e1{\displaystyle e=e^{1}}, shows that en=e×⋯×e⏟n terms{\displaystyle e^{n}=\underbrace {e\times \cdots \times e} _{n{\text{ terms}}}} for positive integers n, relating the exponential function to the elementary notion of exponentiation.

The argument of the exponential function can be any real or complex number, or even an entirely different kind of mathematical object (e.g., matrix).

The ubiquitous occurrence of the exponential function in pure and applied mathematics has led mathematician W. Rudin to opine that the exponential function is “the most important function in mathematics”.[6] In applied settings, exponential functions model a relationship in which a constant change in the independent variable gives the same proportional change (i.e., percentage increase or decrease) in the dependent variable. This occurs widely in the natural and social sciences, as in a self-reproducing population, a fund accruing compound interest, or a growing body of manufacturing expertise. Thus, the exponential function also appears in a variety of contexts within physics, chemistry, engineering, mathematical biology, and economics.

Part of a series of articles on themathematical constant e
Properties
Natural logarithm
Exponential function

Applications
compound interest
Euler’s identity
Euler’s formula
half-lives
exponential growth and decay

Defining e
proof that e is irrational
representations of e
Lindemann–Weierstrass theorem

People
John Napier
Leonhard Euler

Related topics
Schanuel’s conjecture
vte

Formal definition

Main article: Characterizations of the exponential function
The exponential function (in blue), and the sum of the first n + 1 terms of its power series (in red).

The real exponential function exp:R→R{\displaystyle \exp \colon \mathbb {R} \to \mathbb {R} } can be characterized in a variety of equivalent ways. It is commonly defined by the following power series:[6][7]

exp⁡x:=∑k=0∞xkk!=1+x+x22+x36+x424+⋯{\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 }

Since the radius of convergence of this power series is infinite, this definition is, in fact, applicable to all complex numbers z ∈ ℂ (see § Complex plane for the extension of exp⁡x{\displaystyle \exp x} to the complex plane). The constant e can then be defined as e=exp⁡1=∑k=0∞(1/k!).{\textstyle e=\exp 1=\sum _{k=0}^{\infty }(1/k!).}

The term-by-term differentiation of this power series reveals that ddxexp⁡x=exp⁡x{\displaystyle {\frac {d}{dx}}\exp x=\exp x} for all real x, leading to another common characterization of exp⁡x{\displaystyle \exp x} as the unique solution of the differential equation

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

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

Based on this characterization, the chain rule shows that its inverse function, the natural logarithm, satisfies ddyloge⁡y=1/y{\displaystyle {\frac {d}{dy}}\log _{e}y=1/y} for 0,}”>y>0,{\displaystyle y>0,}0,}”> or loge⁡y=∫1y1tdt.{\textstyle \log _{e}y=\int _{1}^{y}{\frac {1}{t}}\,dt.} This relationship leads to a less common definition of the real exponential function exp⁡x{\displaystyle \exp x} as the solution y{\displaystyle y} to the equation

x=∫1y1tdt.{\displaystyle x=\int _{1}^{y}{\frac {1}{t}}\,dt.}

By way of the binomial theorem and the power series definition, the exponential function can also be defined as the following limit:[8][7]

exp⁡x=limn→∞(1+xn)n.{\displaystyle \exp x=\lim _{n\to \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 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[9] to the number

limn→∞(1+1n)n{\displaystyle \lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}}

now known as e. Later, in 1697, Johann Bernoulli studied the calculus of the exponential function.[9]

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,

exp⁡x=limn→∞(1+xn)n{\displaystyle \exp x=\lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}}

first given by Leonhard Euler.[8]
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 the exponential function obeys the basic exponentiation identity,

exp⁡(x+y)=exp⁡xexp⁡y{\displaystyle \exp(x+y)=\exp x\exp y}

which justifies the notation ex for exp x.

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 the function itself (rather than equal to it) is expressible in terms of the exponential function. This function 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.

Derivatives and differential equations

The derivative of the exponential function is equal to the value of the function. From any point P on the curve (blue), let a tangent line (red), and a vertical line (green) with height h be drawn, forming a right triangle with a base b on the x-axis. Since the slope of the red tangent line (the derivative) at P is equal to the ratio of the triangle’s height to the triangle’s base (rise over run), and the derivative is equal to the value of the function, h must be equal to the ratio of h to b. Therefore, the base b must always be 1.

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,

ddxex=exande0=1.{\displaystyle {\frac {d}{dx}}e^{x}=e^{x}\quad {\text{and}}\quad e^{0}=1.}

Functions of the form cex for 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 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 (see Malthusian catastrophe), continuously compounded interest, or radioactive decay—then the variable can be written as a constant times an exponential function of time. Explicitly for any real constant k, a function f: R → R satisfies f′ = kf if and only if f(x) = cekx for some constant c. The constant k is called the decay constant, disintegration constant,[10]rate constant,[11] or transformation constant.[12]

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

ddxef(x)=f′(x)ef(x).{\displaystyle {\frac {d}{dx}}e^{f(x)}=f'(x)e^{f(x)}.}

Continued fractions for ex

A continued fraction for ex can be obtained via an identity of Euler:

ex=1+x1−xx+2−2xx+3−3xx+4−⋱{\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 ez converges more quickly:[13]

ez=1+2z2−z+z26+z210+z214+⋱{\displaystyle e^{z}=1+{\cfrac {2z}{2-z+{\cfrac {z^{2}}{6+{\cfrac {z^{2}}{10+{\cfrac {z^{2}}{14+\ddots }}}}}}}}}

or, by applying the substitution z = x/y:

exy=1+2x2y−x+x26y+x210y+x214y+⋱{\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:

e2=1+40+226+2210+2214+⋱=7+25+17+19+111+⋱{\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, though more slowly, for z > 2. For example:

e3=1+6−1+326+3210+3214+⋱=13+547+914+918+922+⋱{\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 plane

Exponential function on the complex plane. 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. 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:

exp⁡z:=∑k=0∞zkk!{\displaystyle \exp z:=\sum _{k=0}^{\infty }{\frac {z^{k}}{k!}}}

Alternatively, the complex exponential function may defined by modelling the limit definition for real arguments, but with the real variable replaced by a complex one:

exp⁡z:=limn→∞(1+zn)n{\displaystyle \exp z:=\lim _{n\to \infty }\left(1+{\frac {z}{n}}\right)^{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:

exp⁡(w+z)=exp⁡wexp⁡z for all w,z∈C{\displaystyle \exp(w+z)=\exp w\exp z{\text{ for all }}w,z\in \mathbb {C} }

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

exp⁡it=(1−t22!+t44!−t66!+⋯)+i(t−t33!+t55!−t77!+⋯).{\displaystyle \exp it=\left(1-{\frac {t^{2}}{2!}}+{\frac {t^{4}}{4!}}-{\frac {t^{6}}{6!}}+\cdots \right)+i\left(t-{\frac {t^{3}}{3!}}+{\frac {t^{5}}{5!}}-{\frac {t^{7}}{7!}}+\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 exp⁡(±iz){\displaystyle \exp(\pm iz)} and the equivalent power series:[14]

cos⁡z:=exp⁡iz+exp⁡(−iz)2=∑k=0∞(−1)kz2k(2k)!,andsin⁡z:=exp⁡iz−exp⁡(−iz)2i=∑k=0∞(−1)kz2k+1(2k+1)!for all z∈C.{\displaystyle {\begin{aligned}\cos z&:={\frac {\exp iz+\exp(-iz)}{2}}=\sum _{k=0}^{\infty }(-1)^{k}{\frac {z^{2k}}{(2k)!}},\quad {\text{and}}\\sin z&:={\frac {\exp iz-\exp(-iz)}{2i}}=\sum _{k=0}^{\infty }(-1)^{k}{\frac {z^{2k+1}}{(2k+1)!}}\end{aligned}}{\text{for all }}z\in \mathbb {C} .}

The functions exp, cos, and sin so defined have infinite radii of convergence by the ratio test and are therefore entire functions (i.e., holomorphic on C{\displaystyle \mathbb {C} }). The range of the exponential function is C∖{0}{\displaystyle \mathbb {C} \setminus \{0\}}, while the ranges of the complex sine and cosine functions are both C{\displaystyle \mathbb {C} } in its entirety, in accord with Picard’s theorem, which asserts that the range of a nonconstant entire function is either all of C{\displaystyle \mathbb {C} }, or C{\displaystyle \mathbb {C} } excluding one lacunary value.

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

exp⁡iz=cos⁡z+isin⁡z for all z∈C{\displaystyle \exp iz=\cos z+i\sin z{\text{ for all }}z\in \mathbb {C} }.

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

exp⁡z=exp⁡(x+iy):=(exp⁡x)(cos⁡y+isin⁡y){\displaystyle \exp z=\exp(x+iy):=(\exp x)(\cos y+i\sin 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.[15]

For t∈R{\displaystyle t\in \mathbb {R} }, the relationship exp⁡it¯=exp⁡(−it){\displaystyle {\overline {\exp it}}=\exp(-it)} holds, so that |exp⁡it|=1{\displaystyle |\exp it|=1} for real t{\displaystyle t} and t↦exp⁡it{\displaystyle t\mapsto \exp it} maps the real line (mod 2π) to the unit circle in the complex plane. Moreover, going from t=0{\displaystyle t=0} to t=t0{\displaystyle t=t_{0}}, the curve defined by γ(t)=exp⁡it{\displaystyle \gamma (t)=\exp it} traces a segment of the unit circle of length

∫0t0|γ′(t)|dt=∫0t0|iexp⁡it|dt=t0{\displaystyle \int _{0}^{t_{0}}|\gamma ‘(t)|dt=\int _{0}^{t_{0}}|i\exp it|dt=t_{0}},

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 exp⁡(z+2πik)=exp⁡z{\displaystyle \exp(z+2\pi ik)=\exp z} holds for all z∈C,k∈Z{\displaystyle z\in \mathbb {C} ,k\in \mathbb {Z} }.

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

ez+w=ezewe0=1ez≠0ddzez=ez(ez)n=enz,n∈Z for all w,z∈C{\displaystyle {\begin{aligned}e^{z+w}=e^{z}e^{w}\,\e^{0}=1\,\e^{z}\neq 0\{\tfrac {d}{dz}}e^{z}=e^{z}\\left(e^{z}\right)^{n}=e^{nz},n\in \mathbb {Z} \end{aligned}}{\text{ for all }}w,z\in \mathbb {C} }.

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:

zw=ewlog⁡z{\displaystyle z^{w}=e^{w\log 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 zw 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:

(ez)w ≠ ezw, but rather (ez)w = e(z + 2niπ)w multivalued over integers n

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(ex + iy)

z = Im(ex + iy)

z = |ex + iy|

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

v+iw=exp⁡(x+iy){\displaystyle v+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 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:0:\;{\text{green}}}”>x>0:green{\displaystyle x>0:\;{\text{green}}}0:\;{\text{green}}}”>x<0:red{\displaystyle x<0:\;{\text{red}}}0:\;{\text{yellow}}}”>y>0:yellow{\displaystyle y>0:\;{\text{yellow}}}0:\;{\text{yellow}}}”>y<0:blue{\displaystyle y<0:\;{\text{blue}}}

Projection onto the range complex plane (V/W). Compare to the next, perspective picture.

Projection into the x{\displaystyle x}, v{\displaystyle v}, and w{\displaystyle w} dimensions, producing a flared horn or funnel shape (envisioned as 2-D perspective image).

Projection into the y , v, and w dimensions, producing a spiral shape. (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 x axis is mapped to the positive real v axis
the imaginary 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 x axis. It shows the graph is a surface of revolution about the 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 y axis. It shows that the graph’s surface for positive and negative y values doesn’t really meet along the negative real v axis, but instead forms a spiral surface about the y axis. Because its y values have been extended to ±2π, this image also better depicts the 2π periodicity in the imaginary y value.

Computation of ab where both a and b are complex
Main article: Exponentiation

Complex exponentiation ab can be defined by converting a to polar coordinates and using the identity (eln a)b = ab:

ab=(reθi)b=(e(ln⁡r)+θi)b=e((ln⁡r)+θi)b{\displaystyle a^{b}=\left(re^{\theta i}\right)^{b}=\left(e^{(\ln r)+\theta i}\right)^{b}=e^{\left((\ln r)+\theta i\right)b}}

However, when b is not an integer, this function is multivalued, because θ is not unique (see failure of power and logarithm identities).

Matrices and Banach algebras

The power series definition of the exponential function makes sense for square matrices (for which the function is called the matrix exponential) and more generally in any unital Banach algebra B. In this setting, e0 = 1, and ex is invertible with inverse e−x for any x in B. If xy = yx, then ex + y = exey, but this identity can fail for noncommuting x and y.

Some alternative definitions lead to the same function. For instance, ex can be defined as

limn→∞(1+xn)n.{\displaystyle \lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}.}

Or ex can be defined as fx(1), where fx: R→B is the solution to the differential equation dfx/dt(t) = x fx(t), with initial condition fx(0) = 1; it follows that fx(t) = etx for every t in R.

Lie algebras

Given a Lie group G and its associated Lie algebra g{\displaystyle {\mathfrak {g}}}, the exponential map is a map g{\displaystyle {\mathfrak {g}}} ↦ 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 exp(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.

Transcendency

The function ez is not in C(z) (i.e., is not the quotient of two polynomials with complex coefficients).

For n distinct complex numbers {a1, …, an}, the set {ea1z, …, eanz} is linearly independent over C(z).

The function ez is transcendental over C(z).

Computation

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 exp⁡x−1{\displaystyle \exp x-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 ex. For example, if the exponential is computed by using its Taylor series

ex=1+x+x22+x36+⋯+xnn!+⋯,{\displaystyle e^{x}=1+x+{\frac {x^{2}}{2}}+{\frac {x^{3}}{6}}+\cdots +{\frac {x^{n}}{n!}}+\cdots ,}

one may use the Taylor series of ex−1:{\displaystyle e^{x}-1:}

ex−1=x+x22+x36+⋯+xnn!+⋯.{\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 provided by several calculators,[16][17]operating systems (for example Berkeley UNIX 4.3BSD[18]), computer algebra systems, and programming languages (for example C99).[19]

In addition to base e, the IEEE 754-2008 standard defines similar exponential functions near 0 for base 2 and 10: 2x−1{\displaystyle 2^{x}-1} and 10x−1{\displaystyle 10^{x}-1}.

A similar approach has been used for the logarithm (see lnp1).[nb 3]

An identity in terms of the hyperbolic tangent,

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

gives a high-precision value for small values of x on systems that do not implement expm1 x.

See also

Carlitz exponential, a characteristic p analogue
Double exponential function – Exponential function of an exponential function
Exponential field – Mathematical field equipped with an operation satisfying the functional equation of the exponential
Gaussian function
Half-exponential function, a compositional square root of an exponential function
List of exponential topics
List of integrals of exponential functions
Mittag-Leffler function, a generalization of the exponential function
p-adic exponential function
Padé table for exponential function – Padé approximation of exponential function by a fraction of polynomial functions
Tetration – Repeated or iterated exponentiation

Notes

^ In pure mathematics, the notation log x generally refers to the natural logarithm of x or a logarithm in general if the base is immaterial.

^ The notation ln x is the ISO standard and is prevalent in the natural sciences and secondary education (US). However, some mathematicians (e.g., Paul Halmos) have criticized this notation and prefer to use log x for the natural logarithm of x.

^ A similar approach to reduce round-off errors of calculations for certain input values of trigonometric functions consists of using the less common trigonometric functions versine, vercosine, coversine, covercosine, haversine, havercosine, hacoversine, hacovercosine, exsecant and excosecant.

References

^ .mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:”\”””\”””‘””‘”}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background:linear-gradient(transparent,transparent),url(“//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg”)right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:linear-gradient(transparent,transparent),url(“//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg”)right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background:linear-gradient(transparent,transparent),url(“//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg”)right 0.1em center/9px no-repeat}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:linear-gradient(transparent,transparent),url(“//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg”)right 0.1em center/12px no-repeat}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}”Compendium of Mathematical Symbols”. Math Vault. 2020-03-01. Retrieved 2020-08-28.

^ Goldstein, Larry Joel; Lay, David C.; Schneider, David I.; Asmar, Nakhle H. (2006). Brief calculus and its applications (11th ed.). Prentice–Hall. ISBN 978-0-13-191965-5. (467 pages)

^ Courant; Robbins (1996). Stewart (ed.). What is Mathematics? An Elementary Approach to Ideas and Methods (2nd revised ed.). Oxford University Press. p. 448. ISBN 978-0-13-191965-5. This natural exponential function is identical with its derivative. This is really the source of all the properties of the exponential function, and the basic reason for its importance in applications…

^ “Exponential Function Reference”. www.mathsisfun.com. Retrieved 2020-08-28.

^ Converse, Henry Augustus; Durell, Fletcher (1911). Plane and Spherical Trigonometry. Durell’s mathematical series. C. E. Merrill Company. p. 12. Inverse Use of a Table of Logarithms; that is, given a logarithm, to find the number corresponding to it, (called its antilogarithm) … [1]

^ a b Rudin, Walter (1987). Real and complex analysis (3rd ed.). New York: McGraw-Hill. p. 1. ISBN 978-0-07-054234-1.

^ a b Weisstein, Eric W. “Exponential Function”. mathworld.wolfram.com. Retrieved 2020-08-28.

^ a b Maor, Eli. e: the Story of a Number. p. 156.

^ a b O’Connor, John J.; Robertson, Edmund F. (September 2001). “The number e”. School of Mathematics and Statistics. University of St Andrews, Scotland. Retrieved 2011-06-13.

^ Serway (1989, p. 384) harvtxt error: no target: CITEREFSerway1989 (help)

^ Simmons (1972, p. 15)

^ McGraw-Hill (2007)

^ Lorentzen, L.; Waadeland, H. (2008). “A.2.2 The exponential function.”. Continued Fractions. Atlantis Studies in Mathematics. 1. p. 268. doi:10.2991/978-94-91216-37-4. ISBN 978-94-91216-37-4.

^ Rudin, Walter (1976). Principles of Mathematical Analysis. New York: McGraw-Hill. p. 182. ISBN 978-0-07054235-8.

^ Apostol, Tom M. (1974). Mathematical Analysis (2nd ed.). Reading, Mass.: Addison Wesley. pp. 19. ISBN 978-0-20100288-1.

^ HP 48G Series – Advanced User’s Reference Manual (AUR) (4 ed.). Hewlett-Packard. December 1994 [1993]. HP 00048-90136, 0-88698-01574-2. Retrieved 2015-09-06.

^ HP 50g / 49g+ / 48gII graphing calculator advanced user’s reference manual (AUR) (2 ed.). Hewlett-Packard. 2009-07-14 [2005]. HP F2228-90010. Retrieved 2015-10-10. [2]

^ Beebe, Nelson H. F. (2017-08-22). “Chapter 10.2. Exponential near zero”. The Mathematical-Function Computation Handbook – Programming Using the MathCW Portable Software Library (1 ed.). Salt Lake City, UT, USA: Springer International Publishing AG. pp. 273–282. doi:10.1007/978-3-319-64110-2. ISBN 978-3-319-64109-6. LCCN 2017947446. S2CID 30244721. Berkeley UNIX 4.3BSD introduced the expm1() function in 1987.

^ Beebe, Nelson H. F. (2002-07-09). “Computation of expm1 = exp(x)−1” (PDF). 1.00. Salt Lake City, Utah, USA: Department of Mathematics, Center for Scientific Computing, University of Utah. Retrieved 2015-11-02.

McGraw-Hill Encyclopedia of Science & Technology (10th ed.). New York: McGraw-Hill. 2007. ISBN 978-0-07-144143-8.
Serway, Raymond A.; Moses, Clement J.; Moyer, Curt A. (1989), Modern Physics, Fort Worth: Harcourt Brace Jovanovich, ISBN 0-03-004844-3
Simmons, George F. (1972), Differential Equations with Applications and Historical Notes, New York: McGraw-Hill, LCCN 75173716

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Exponential Decay Functions Flashcards | Quizlet

Exponential Decay Functions Flashcards | Quizlet – Start studying Exponential Decay Functions. Learn vocabulary, terms and more with flashcards, games and other study tools. What is the multiplicative rate of change of the function?WildstyleSmart WildstyleSmart. I think its 2/5 = .4 because of the negative in the exponent, it is 0.4 Hope I helpedFor this exponential equation, we expect a negative slope/average rate of change, because the negative sign in the exponent indicates we have an exponential decay curve. The slope/average rate of change between any two points will be negative.

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Rules of differentiation – constant, power, constant multiple, sum rule – When we've talked about derivatives so far, we've talked about them in a very formal and time consuming way.
For instance, the instantaneous rate of change, we can think of the slope of the tangent line. And one way to write
that is the slope of the tangent line is equal to the limit as x approaches a of f of x minus f of a divided by x minus a. Or, we could write it as the limit as h approaches zero of f
of a plus h minus f of a divided by h. We could also call that f prime of a, the first
derivative at a. We've also talked about the derivative as being a function. So instead
of at a point a, we can say f prime of x is equal to the limit as h approaches zero of
f of x plus h minus f of x all over h. Well, if we had to do this every single time we
found a derivative, … I love math, but I don't love math this much. So we are going to come up with some general rules that you'll need to memorize. Let's first talk about a
constant. That is, where f of x equals a number c. And c is going to be a real number. Remember, the derivative is the rate of change of the function. If I have a constant function, there is no rate of change. Therefore, the rate of change is equal to zero. This leads us
to our first rule. The Constant Rule. If I have the derivative with respect to x of c,
that equals zero if c is a real number. The next rule we're going to talk about is the
Power Rule. Before I prove it, and I'm not going to do a lot of proofs, but I am going
to do this first proof. But the Power Rule is if I take the derivative of a variable
x raised to the nth power, that is equal to n times x to the n minus one power. And let's prove this one. The first thing I am going to do is take my formal definition and start off with taking the derivative of x. That is, x to the first power. So if I let n equal
1, that means f of x is equal to x to the first power, or just x. And if I go ahead
and use my definition of f prime of a, then I find that I have the limit as x approaches
a of x minus a divided by x minus a which is simply equal to 1. Which, by the way, does match my power rule. Let's go ahead and check that. And this looks fine. The n is one, so
the derivative of x to the first power is 1 times x to the 1 minus 1, which is x to
the zero power. Anything to the zero power is 1, therefore my power rule does know that this would, in fact, equal 1. So now let's say n is going to be greater than or equal
to 2. And f of x is equal to x to the n power. Now we can say the first derivative of f of
a is equal to the limit of x approaching a of x to the n minus a to the n, divided by
x minus a. There is a factoring rule that we can always factor this x to the n minus
a to the n into this. So if I can factor x to the n minus a to the n in this form, I
see that, first of all, I left off the limit. Let me fix that. There we go. And now we see this x minus a divides out of the numerator and the denominator, and I'm left with the
following. Once I have this in this form, I can go ahead and directly substitute x as a, because this is a polynomial (we remember our limit laws). If I go ahead and multiply
this all out, I have a bunch of just a to the n minus ones. In fact, it's not just a
bunch of a to the n minus ones, there's exactly n of them. And so I get n time a to the n
minus 1. If I made this a function, I would find that f prime of x is equal to n times
x to the n minus one. Which is what I have for the power rule. This is the only one that I am going to go through and do a proof of, you don't have to recreate the proof, but
you do have to be able to use the power rule. Notice that this power rule will actually
work in the constants case.That is, if I had a constant, say, c, it would be c times x
to the zero. By the power rule, that would be zero times x to the zero minus one, but
anything times zero is zero. So the constant rule is really rolled into the power rule.
By the power rule, the derivative of x with respect to x of x to the fifth, that's simply
equal to five times x to the five minus one, or, 5 times x to the fourth. The second one – it would be really tempting to say the derivative with respect to x of 3 to the sixth is 6 times 3 to the fifth power, but of course that's not right because 3 to the sixth is actually
a constant. There's no x in there. So this is still equal to zero. Don't fall in that
trap. Our next rule is going to be the Constant Multiple Rule. That is, if I take the derivative of a constant c times a function f of x, that is simply equal to c times the derivative
of f of x. I've got two examples up here. So my first one is, the derivative with respect to x of negative 5 sixths, x to the tenth power. So the first thing I will do is pull
out that constant. So I have negative five sixths d dx x to the tenth. Now I am going
to use my power rule. And that gives me negative 5 sixths times ten times x to the ninth power. And if I simplify this, I'll get the following. Negative 25 over 3 x to the ninth power. Now my second example, I'm going to first again use the constant multiple rule to pull out
that 1 fifteenth. Notice now I'm taking the derivative of with respect to t. It works
the same way. Whatever my independent variable is. So, I've rewritten the square root of
t as t to the one half power, because those are the same thing. And although I didn't
specify, I'm going to go back and say with my power rule, that n has to just be a real
number. This I didn't prove, and Icould, but I'm just not going to. So I am going to say
that, using that same power rule, I'm going to get one over 15 times one half, that's
my exponent, times t to the one half minus one power. Or, 1 over 15 times one half time t to the negative one half power. I can rewrite this as such. Generally, if I have started
off with giving you information in square root form, I am going to rewrite it as a square root and t to the negative one half power, that's just equal to one over the square root of t. And that's my final answer. The next rule is the sum rule. That is, the derivative
with respect to x of f of x plus g of x is simply equal to the derivatives of the separate functions added together. I am warning you – do not assume that this is going to work
with products. That is, with multiplication or division. But right now, we are just talking about addition and subtraction. And that is very straightforward. Let's do a quick example. The derivative with respect to x of all of this is equal to the derivative of their individual terms. Again, this is by the sum rule. When you're taking derivatives, you're not going
to have to write out each rule every time like we did with the limit rules, however,
I want to specify this as I am teaching it so you can understand what I am doing, step by step. The next rule I am going to use is the constant multiple rule. And that allows me to pull the constants out. Finally, I am going to use my power and constant rules to come up with the following. And finally, this is what I get as my answer. There's one more special derivative we're going to talk about. And that's the derivative of e to the x. What is e? e is equal to an irrational number, 2 point 718 and a whole bunch of other digits. And what makes e special is this – that is, if I take e, and raise it to the h power,
subtract 1 from it, and divide by h, and let the limit of h approach zero, this is equal
to the number 1. And here's a graph of e to the x, and the slope at zero is actually equal to one. So where does this get us? So the derivative of e to the x is equal to the limit
as h approaches 0 of e to the x plus h minus e to the x divided by h. And again, that is
by the definition of the derivative. So let's go ahead and rewrite my exponent as such. I see both of my terms has an e to the x in them. So I am going to factor that out. And now that I factored out that e to the x, I realize the limit as h approaches zero doesn't affect the e to the x at all. So I can go ahead and pull that out. Well, I've already
said that the limit of h approaching zero of e to the h minus 1 over h is equal to the
number 1. So that means that the derivative of e to the x is simply e to the x. It's the
BEST derivative out there. And again, I'll write it out. The derivative of e to the x
is simply e to the x. One final thing to talk about — I can take derivatives higher than
just the first derivative. I could take, for instance, the second derivative. The second derivative of f of x is simply the derivative with respect to x of the first derivative
of f of x. And I can expand that to talk about any values of n. And the one thing I want
to point out is if I talk about the nth derivative, I put that n in parentheses. So that f to
the parentheses n is not f to the nth power. It's the nth derivative of f. So that just
equals the derivative of the f to the n minus 1nth derivative of x. And that's our first
round of rules of differentiation. .