Exponential function

Formal definition

The exponential function exp : C → C can be characterized in a variety of equivalent ways. Most commonly, it is defined by the following power series:

Since the radius of convergence of this power series is infinite, this definition is applicable to all complex numbers z . The constant e is then defined as e = exp ⁡ ( 1 ) = ∑ k = 0 ∞ ( 1 / k ! ) .

Less commonly, the real exponential function is defined as the solution y to the equation

The exponential function can also be defined as the following limit:

Overview

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

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,

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,

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 importance of the exponential function in mathematics and the sciences stems mainly from properties of its derivative. In particular,

Proof:

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:

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:

Continued fractions for ex

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

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

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

with a special case for z = 2:

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

Complex plane

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 ∞ z k k ! .

Termwise multiplication of two copies of these 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 ⁡ ( w ) exp ⁡ ( z ) for all w , z ∈ 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 = i t ( t real), the series definition yields the expansion

exp ⁡ ( i t ) = ( 1 − t 2 2 ! + t 4 4 ! − t 6 6 ! + ⋯ ) + i ( t − t 3 3 ! + t 5 5 ! − t 7 7 ! + ⋯ ) .

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 ⁡ ( ± i z ) and the equivalent power series:

cos ⁡ z := 1 2 [ exp ⁡ ( i z ) + exp ⁡ ( − i z ) ] = ∑ k = 0 ∞ ( − 1 ) k z 2 k ( 2 k ) ! and sin ⁡ z := 1 2 i [ exp ⁡ ( i z ) − exp ⁡ ( − i z ) ] = ∑ k = 0 ∞ ( − 1 ) k z 2 k + 1 ( 2 k + 1 ) ! , for all z ∈ 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 ). The range of the exponential function is C ∖ { 0 } , while the ranges of the complex sine and cosine functions are both 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 , or C excluding one lacunary value.

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

exp ⁡ ( i z ) = cos ⁡ z + i sin ⁡ z for all z ∈ C .

We could alternatively define the complex exponential function based on this relationship. If z = x + i y , where x and y are both real, then we could define its exponential as

exp ⁡ z = exp ⁡ ( x + i y ) := ( 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.

For t ∈ R , the relationship exp ⁡ ( i t ) ¯ = exp ⁡ ( − i t ) holds, so that | exp ⁡ ( i t ) | = 1 for real t and t ↦ exp ⁡ ( i t ) maps the real line (mod 2 π ) to the unit circle. Based on the relationship between exp ⁡ ( i t ) and the unit circle, it is easy to see that, restricted to real arguments, the definitions of sine and cosine given above coincide with their more elementary definitions based on geometric notions.

The complex exponential function is periodic with period 2 π i and exp ⁡ ( z + 2 π i k ) = exp ⁡ z holds for all z ∈ C , k ∈ Z .

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

for all w , z ∈ 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:

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:

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.

Computation of ab where both a and b are complex

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

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 g , the exponential map is a map 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 ez

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.