exponential function

指数関数

関

exponential

WordNet

1. perform as expected when applied; "The washing machine wont go unless its plugged in"; "Does this old car still run well?"; "This old radio doesnt work anymore" (同)work, operate, go, run
2. (mathematics) a mathematical relation such that each element of a given set (the domain of the function) is associated with an element of another set (the range of the function) (同)mathematical function, single-valued function, map, mapping
3. a relation such that one thing is dependent on another; "height is a function of age"; "price is a function of supply and demand"
4. the actions and activities assigned to or required or expected of a person or group; "the function of a teacher"; "the government must do its part"; "play its role" (同)office, part, role
5. what something is used for; "the function of an auger is to bore holes"; "ballet is beautiful but what use is it?" (同)purpose, role, use
6. a formal or official social gathering or ceremony; "it was a black-tie function"
7. of or involving exponents; "exponential growth"
8. a function in which an independent variable appears as an exponent (同)exponential_function
9. performing or able to perform its regular function; "a functioning flashlight"

PrepTutorEJDIC

1. (人・物の果たすべき)『働き』,機能,役目;《しばしば複数形で》職務 / (公式の)儀式,《話》大きな催し(集まり),式典 / (数学で)関数 / 〈機械などが〉『機能を果たす』,動く / (…の)役目をする,代用となる《+『as』+『名』》
2. (数学で)指数の

Wikipedia preview

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2016/10/08 02:57:43」(JST)

wiki en

[Wiki en表示]

This article is about functions of the form f(x) = ab^x. For functions of the form f(x,y) = x^y, see Exponentiation. For functions of the form f(x) = x^r, see Power function.

The natural exponential function y = e^x

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

$f(x)=b^{x}\,$

in which the input variable x occurs as an exponent. A function of the form f(x) = b^{x + c} is also considered an exponential function, and a function of the form f(x) = a·b^x can be re-written as f(x) = b^{x + c} by the use of logarithms and so is an exponential function.

Exponential functions are uniquely characterized by the fact that the growth rate of such a function is directly proportional to the value of the function. This proportionality can be expressed by saying

${\frac {d}{dx}}b^{x}={\frac {d}{dx}}\exp(x\ln b)$

$=\exp(x\ln b)\cdot {\frac {d}{dx}}(x\ln b)$

$=b^{x}\ln b$

where ln b is a constant, and a constant is a quantity that does not change as the variable x changes.

For just one base b this constant factor is equal to 1, and that base is the number e ≈ 2.71828...:

${\frac {d}{dx}}e^{x}=e^{x}\times 1$

This equality makes it possible to reduce some questions in mathematical analysis of exponential functions to the analysis of this one exponential function, conventionally called the "natural exponential function" and denoted by

$x\mapsto \exp(x).\,$

The exponential function models 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. The function is often written as exp(x), especially when it is impractical to write the independent variable as a superscript. The exponential function is widely used in physics, chemistry, engineering, mathematical biology, economics and mathematics.

Exponential function

Representation	ex
Inverse	ln x
Derivative	ex
Indefinite Integral	ex + C

The graph of y = e^x is upward-sloping, and increases faster as x increases. The graph always lies above the x-axis but can get arbitrarily close to it for negative x; thus, the x-axis is a horizontal asymptote. The slope of the tangent to the graph at each point is equal to its y-coordinate at that point. The inverse function is the natural logarithm ln(x); because of this, some old texts refer to the exponential function as the antilogarithm.

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.

Part of a series of articles on the
mathematical 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

Formal definition

Main article: Characterizations of the exponential function

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

The exponential function e^x can be characterized in a variety of equivalent ways. In particular it may be defined by the following power series:

$e^{x}=\sum _{n=0}^{\infty }{x^{n} \over n!}=1+x+{x^{2} \over 2!}+{x^{3} \over 3!}+{x^{4} \over 4!}+\cdots$

Using an alternate definition for the exponential function leads to the same result when expanded as a Taylor series.

Less commonly, e^x is defined as the solution y to the equation

$x=\int _{1}^{y}{1 \over t}\mathrm {d} t$

It is also the following limit:

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

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 that led Jacob Bernoulli in 1683 to the number

$\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}$

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

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,

$\exp(x)=\lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}$

first given by 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 the exponential function obeys the basic exponentiation identity,

$\exp(x+y)=\exp(x)\cdot \exp(y)$

which is why it can be written as e^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 and 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 properties of its derivative. In particular,

${\frac {\mathrm {d} }{\mathrm {d} x}}e^{x}=e^{x}$

Proof:

${\begin{aligned}e^{x}&=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+{\frac {x^{5}}{5!}}+\cdots \\{\mathrm {d} \over \mathrm {d} x}e^{x}&={\mathrm {d} \over \mathrm {d} x}\left(1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+{\frac {x^{5}}{5!}}+\cdots \right)\\&=0+1+{\frac {2x}{2!}}+{\frac {3x^{2}}{3!}}+{\frac {4x^{3}}{4!}}+{\frac {5x^{4}}{5!}}+\cdots \\&=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+{\frac {x^{5}}{5!}}+\cdots \\&=e^{x}\\\end{aligned}}$

That is, e^x is its own derivative and hence is a simple example of a Pfaffian function. Functions of the form ce^x for constant c are the only functions with that property (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) = ce^kx for some constant c.

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

${\mathrm {d} \over \mathrm {d} x}e^{f(x)}=f'(x)e^{f(x)}$

Continued fractions for e^x

A continued fraction for e^x can be obtained via an identity of Euler:

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

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

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

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

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

with a special case for z = 2:

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

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

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

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. One such definition parallels the power series definition for real numbers, where the real variable is replaced by a complex one:

$e^{z}=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}$

The exponential function is periodic with imaginary period 2πi and can be written as

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

where a and b are real values and on the right the real functions must be used if used as a definition (see also Euler's formula). This formula connects the exponential function with the trigonometric functions and to the hyperbolic functions.

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

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

for all z and w.

The exponential function is an entire function as it is holomorphic over the whole complex plane. It takes on every complex number excepting 0 as value; that is, 0 is a lacunary value of the exponential function. This is an example of Picard's little theorem that any non-constant entire function takes on every complex number as value with at most one value excepted.

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:

$z^{w}=e^{w\log z}$

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 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
≠ e^zw, but rather (e^z)w
= e^{(z + 2πin)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 might be noted: 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.

• 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 a^b where both a and b are complex

Main article: Exponentiation

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

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

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 Banach algebra B. In this setting, e⁰ = 1, and e^x is invertible with inverse e^−x for any x in B. If xy = yx, then e^{x + y} = e^xe^y, but this identity can fail for noncommuting x and y.

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

$\lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}.$

Or e^x can be defined as f(1), where f: R→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 ${\mathfrak {g}}$, the exponential map is a map ${\mathfrak {g}}$ ↦ 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.

Similar properties of e and the function e^z

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

For n distinct complex numbers {a₁, …, a_n}, the set {e^a₁z, …, e^a_nz} is linearly independent over C(z).

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

exp and expm1

Some calculators provide a dedicated exp(x) function designed to provide a higher precision than achievable by using e^x directly.

Based on a proposal by William Kahan and first implemented in the Hewlett-Packard HP-41C calculator in 1979, some scientific calculators, computer algebra systems and programming languages (for example C99) support a special exponential minus 1 function alternatively named E^X-1, expm1(x), expm(x), or exp1m(x) to provide more accurate results for values of x near zero compared to using exp(x)-1 directly. This function is implemented using a different internal algorithm to avoid an intermediate result near 1, thereby allowing both the argument and the result to be near zero. Similar inverse functions named lnp1(x), ln1p(x) or log1p(x) exist as well.

See also

• Approximating natural exponents (log base e)
• Carlitz exponential, a characteristic p analogue
• Characterizations of the exponential function
• Double exponential function
• e (mathematical constant)
• Exponential decay
• Exponential field
• Exponential growth
• Exponentiation
• Half-exponential function, a compositional square root of an exponential function
• Linear function
• List of exponential topics
• List of integrals of exponential functions
• Mittag-Leffler function, a generalization of the exponential function
• p-adic exponential function
• Polynomial function
• Padé table for exponential function – Padé approximation if exponential function by a fraction of polynomial functions
• Tetration

Notes

References

External links

• Hazewinkel, Michiel, ed. (2001), "Exponential function", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 

"Complex exponential function". PlanetMath. 

"Derivative of exponential function". PlanetMath. 

• Derivative of exponential function interactive graph
• Weisstein, Eric W. "Exponential Function". MathWorld. 

• Taylor Series Expansions of Exponential Functions at efunda.com
• Complex exponential interactive graphic
• Derivative of exp(xⁿ) by limit definition
• General exponential limit

UpToDate Contents

全文を閲覧するには購読必要です。 To read the full text you will need to subscribe.

• 1. 肺機能検査の基準値 reference values for pulmonary function testing
• 2. 腎機能の評価 assessment of kidney function
• 3. 先天性および後天性血小板機能障害 congenital and acquired disorders of platelet function
• 4. 健常女性における卵巣の発達および機能不全（閉経） ovarian development and failure menopause in normal women
• 5. 膵外分泌機能検査 pancreatic exocrine function tests

English Journal

• Inactivation of microorganisms by low-frequency high-power ultrasound: 1. Effect of growth phase and capsule properties of the bacteria.

• Gao S, Lewis GD, Ashokkumar M, Hemar Y.SourceSchool of Chemical Sciences, The University of Auckland, Private Bag 92019, Auckland, New Zealand; School of Biological Sciences, The University of Auckland, Private Bag 92019, Auckland, New Zealand; Institute of Food and Agricultural Standardization, China National Institute of Standardization, Beijing 10088, China.
• Ultrasonics sonochemistry.Ultrason Sonochem.2014 Jan;21(1):446-53. doi: 10.1016/j.ultsonch.2013.06.006. Epub 2013 Jun 20.
• The aim of this study was to determine the effects of high-intensity low-frequency (20kHz) ultrasound treatment on the viability of bacteria suspension. More specifically, we have investigated the relationship between the deactivation efficiency and the physical (size, hydrophobicity) and biological
• PMID 23835398

• Oxidized albumin. The long way of a protein of uncertain function.

• Bruschi M, Candiano G, Santucci L, Ghiggeri GM.SourceLaboratory on Pathophysiology of Uremia, Istituto Giannina Gaslini, Genoa, Italy.
• Biochimica et biophysica acta.Biochim Biophys Acta.2013 Dec;1830(12):5473-9. doi: 10.1016/j.bbagen.2013.04.017. Epub 2013 Apr 22.
• BACKGROUND: Proteins are extremely reactive to oxidants and should represent a potential target of instable reactive oxygen. This may represent a problem for plasma proteins since they may be directly modified in vivo in a compartment where antioxidant enzymatic systems are scarcely represented. On
• PMID 23618696

• Activated effect of lignin on α-amylase.

• Zhang J, Cui JH, Yin T, Sun L, Li G.SourceSchool of Life Sciences, Shanghai University, Shanghai 200444, PR China.
• Food chemistry.Food Chem.2013 Dec 1;141(3):2229-37. doi: 10.1016/j.foodchem.2013.05.047. Epub 2013 May 23.
• This paper reports a new kind of activator of α-amylase, lignin, which can greatly increase α-amylase activity. The promoted ratio of lignin is even much higher than that of chloride ion, the traditional activator of α-amylase. Further experimental results reveal that lignin may interact with α-
• PMID 23870952

Japanese Journal

• イオン液体中での溶媒和ダイナミクスの励起波長依存性の研究

• 木村 佳文,澁谷 真子,八坂 能郎 [他],上野 正勝,キムラ ヨシフミ,シブヤ マコ,ヤサカ ヨシロウ,ウエノ マサカツ,Kimura Yoshifumi,Shibuya Mako,Yasaka Yoshiro,Ueno Masakatsu
• 同志社大学理工学研究報告 55(4), 328-334, 2015-01
• … The peak position of the fluorescence spectrum at each delay time was evaluated by simulating the spectrum by a log-normal function. … The time profile of the peak position was well simulated by a stretched-exponential function. … The average relaxation times evaluated by the integration of the relaxation function were weakly dependent on the excitation wavelength, and they were correlated with the viscosity of the solvent. …
• NAID 120005550053

• 指数関数型非線形構成則に従う一次元棒における応力波伝播挙動の厳密な解析

• 渡辺 知規
• 日本機械学会論文集 81(821), 14-00265-14-00265, 2015
• … An exponential function is chosen as the constitutive equation in order to realize that the one-dimensional model possesses some useful properties for the analysis. …
• NAID 130005005004

• Robustness of noninvertible dichotomies

• ,
• Journal of the Mathematical Society of Japan 67(1), 293-317, 2015
• … We establish the robustness of exponential dichotomies for evolution families of linear operators in a Banach space, in the sense that the existence of an exponential dichotomy persists under sufficiently small linear perturbations. … Moreover, we consider the general cases of nonuniform exponential dichotomies and of dichotomies that may exhibit stable and unstable behaviors with respect to arbitrary asymptotic rates ecρ(t) for some function ρ(t). …
• NAID 130004927758

Related Pictures

---

★リンクテーブル★

リンク元	「指数関数」
関連記事	「function」「exponential」

「指数関数」

　　[★]

英

exponential function、exponential

関

指数関数的

「function」

　　[★]

• n.

• 機能、(数学)関数、(化学)官能基

survival as a function of HbA1c in people with type 2 diabetes HbA1cの機能を考慮（指標に）した2型糖尿病の患者の生存（予後）

Figure 2A shows survival as a function of device removal during index hpspitalization, censored at 1 year or date of last contact. (Figure2Aはは最初の入院中における心臓デバイス除去に関する生存曲線で退院後1年もしくは最後の接触で打ち切っている)

• v.

• 機能する

関

functional、functional group、functionalise、functionalize、operate

　 　 　 　 　 　 　

「exponential」

　　[★]

• adj.

• 指数関数的な

• n.

• 指数関数

関

exponential function、exponentially