# Exponential function

For functions of the form f(x,y) = x, see Exponentiation. Exponential function_sentence_1

For functions of the form f(x) = x, see Power function. Exponential function_sentence_2

In mathematics, an exponential function is a function of the form Exponential function_sentence_3

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. Exponential function_sentence_4

The constant of proportionality of this relationship is the natural logarithm of the base b: Exponential function_sentence_5

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: Exponential function_sentence_6

The exponential function satisfies the fundamental multiplicative identity (which can be extended to complex-valued exponents as well): Exponential function_sentence_7

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). Exponential function_sentence_8

The ubiquitous occurrence of the exponential function in pure and applied mathematics has led mathematician W. Exponential function_sentence_9 Rudin to opine that the exponential function is "the most important function in mathematics". Exponential function_sentence_10

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. Exponential function_sentence_11

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. Exponential function_sentence_12

Thus, the exponential function also appears in a variety of contexts within physics, chemistry, engineering, mathematical biology, and economics. Exponential function_sentence_13

## Formal definition Exponential function_section_0

Main article: Characterizations of the exponential function Exponential function_sentence_14

By way of the binomial theorem and the power series definition, the exponential function can also be defined as the following limit: Exponential function_sentence_15

## Overview Exponential function_section_1

The exponential function arises whenever a quantity grows or decays at a rate proportional to its current value. Exponential function_sentence_16

One such situation is continuously compounded interest, and in fact it was this observation that led Jacob Bernoulli in 1683 to the number Exponential function_sentence_17

now known as e. Later, in 1697, Johann Bernoulli studied the calculus of the exponential function. Exponential function_sentence_18

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). Exponential function_sentence_19

If instead interest is compounded daily, this becomes (1 + x/365). Exponential function_sentence_20

Letting the number of time intervals per year grow without bound leads to the limit definition of the exponential function, Exponential function_sentence_21

first given by Leonhard Euler. Exponential function_sentence_22

This is one of a number of characterizations of the exponential function; others involve series or differential equations. Exponential function_sentence_23

From any of these definitions it can be shown that the exponential function obeys the basic exponentiation identity, Exponential function_sentence_24

which justifies the notation e for exp x. Exponential function_sentence_25

The derivative (rate of change) of the exponential function is the exponential function itself. Exponential function_sentence_26

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. Exponential function_sentence_27

This function property leads to exponential growth or exponential decay. Exponential function_sentence_28

The exponential function extends to an entire function on the complex plane. Exponential function_sentence_29

Euler's formula relates its values at purely imaginary arguments to trigonometric functions. Exponential function_sentence_30

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. Exponential function_sentence_31

## Derivatives and differential equations Exponential function_section_2

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. Exponential function_sentence_32

That is, Exponential function_sentence_33

Functions of the form ce for constant c are the only functions that are equal to their derivative (by the Picard–Lindelöf theorem). Exponential function_sentence_34

Other ways of saying the same thing include: Exponential function_sentence_35

Exponential function_unordered_list_0

• The slope of the graph at any point is the height of the function at that point.Exponential function_item_0_0
• The rate of increase of the function at x is equal to the value of the function at x.Exponential function_item_0_1
• The function solves the differential equation y′ = y.Exponential function_item_0_2
• exp is a fixed point of derivative as a functional.Exponential function_item_0_3

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. Exponential function_sentence_36

Explicitly for any real constant k, a function f: R → R satisfies f′ = kf if and only if f(x) = ce for some constant c. The constant k is called the decay constant, disintegration constant, rate constant, or transformation constant. Exponential function_sentence_37

Furthermore, for any differentiable function f(x), we find, by the chain rule: Exponential function_sentence_38

## Continued fractions for ex Exponential function_section_3

A continued fraction for e can be obtained via an identity of Euler: Exponential function_sentence_39

The following generalized continued fraction for e converges more quickly: Exponential function_sentence_40

or, by applying the substitution z = x/y: Exponential function_sentence_41

with a special case for z = 2: Exponential function_sentence_42

This formula also converges, though more slowly, for z > 2. Exponential function_sentence_43

For example: Exponential function_sentence_44

## Complex plane Exponential function_section_4

As in the real case, the exponential function can be defined on the complex plane in several equivalent forms. Exponential function_sentence_45

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: Exponential function_sentence_46

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: Exponential function_sentence_47

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: Exponential function_sentence_48

The definition of the complex exponential function in turn leads to the appropriate definitions extending the trigonometric functions to complex arguments. Exponential function_sentence_49

In this expansion, the rearrangement of the terms into real and imaginary parts is justified by the absolute convergence of the series. Exponential function_sentence_50

The real and imaginary parts of the above expression in fact correspond to the series expansions of cos t and sin t, respectively. Exponential function_sentence_51

These definitions for the exponential and trigonometric functions lead trivially to Euler's formula: Exponential function_sentence_52

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. Exponential function_sentence_53

When its domain is extended from the real line to the complex plane, the exponential function retains the following properties: Exponential function_sentence_54

Extending the natural logarithm to complex arguments yields the complex logarithm log z, which is a multivalued function. Exponential function_sentence_55

We can then define a more general exponentiation: Exponential function_sentence_56

for all complex numbers z and w. This is also a multivalued function, even when z is real. Exponential function_sentence_57

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. Exponential function_sentence_58

The rule about multiplying exponents for the case of positive real numbers must be modified in a multivalued context: Exponential function_sentence_59

Exponential function_description_list_1

• (e) ≠ e, but rather (e) = e multivalued over integers nExponential function_item_1_4

See failure of power and logarithm identities for more about problems with combining powers. Exponential function_sentence_60

The exponential function maps any line in the complex plane to a logarithmic spiral in the complex plane with the center at the origin. Exponential function_sentence_61

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. Exponential function_sentence_62

Exponential function_unordered_list_2

• 3D-Plots of Real Part, Imaginary Part, and Modulus of the exponential functionExponential function_item_2_5
• Exponential function_item_2_6
• Exponential function_item_2_7
• Exponential function_item_2_8

Considering the complex exponential function as a function involving four real variables: Exponential function_sentence_63

the graph of the exponential function is a two-dimensional surface curving through four dimensions. Exponential function_sentence_64

Exponential function_unordered_list_3

• Graphs of the complex exponential functionExponential function_item_3_9
• Exponential function_item_3_10
• Exponential function_item_3_11
• Exponential function_item_3_12
• Exponential function_item_3_13

The second image shows how the domain complex plane is mapped into the range complex plane: Exponential function_sentence_65

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. Exponential function_sentence_66

### Computation of ab where both a and b are complex Exponential function_section_5

Main article: Exponentiation Exponential function_sentence_67

Complex exponentiation a can be defined by converting a to polar coordinates and using the identity (e) = a: Exponential function_sentence_68

However, when b is not an integer, this function is multivalued, because θ is not unique (see failure of power and logarithm identities). Exponential function_sentence_69

## Matrices and Banach algebras Exponential function_section_6

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. Exponential function_sentence_70

In this setting, e = 1, and e is invertible with inverse e for any x in B. Exponential function_sentence_71

If xy = yx, then e = ee, but this identity can fail for noncommuting x and y. Exponential function_sentence_72

Some alternative definitions lead to the same function. Exponential function_sentence_73

For instance, e can be defined as Exponential function_sentence_74

Or e 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) = e for every t in R. Exponential function_sentence_75

## Lie algebras Exponential function_section_7

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. Exponential function_sentence_76

## Transcendency Exponential function_section_8

The function e is not in C(z) (i.e., is not the quotient of two polynomials with complex coefficients). Exponential function_sentence_77

For n distinct complex numbers {a1, …, an}, the set {e, …, e} is linearly independent over C(z). Exponential function_sentence_78

The function e is transcendental over C(z). Exponential function_sentence_79

## Computation Exponential function_section_9

Following a proposal by William Kahan, it may thus be useful to have a dedicated routine, often called expm1, for computing e − 1 directly, bypassing computation of e. For example, if the exponential is computed by using its Taylor series Exponential function_sentence_80

This was first implemented in 1979 in the Hewlett-Packard HP-41C calculator, and provided by several calculators, operating systems (for example Berkeley UNIX 4.3BSD), computer algebra systems, and programming languages (for example C99). Exponential function_sentence_81

A similar approach has been used for the logarithm (see lnp1). Exponential function_sentence_82

An identity in terms of the hyperbolic tangent, Exponential function_sentence_83

gives a high-precision value for small values of x on systems that do not implement expm1(x). Exponential function_sentence_84