Exponential function - Encyclopedia of Mathematics (2024)

exponent

The function

$$ y = e ^ {z} \equiv \mathop{\rm exp} z ,$$

where $ e $is the base of the natural logarithm, which is also known as the Napier number. This function is defined for any value of $ z $ (real or complex) by

$$ \tag{1 }e ^ {z} = \lim\limits _ {n \rightarrow \infty } \left ( 1 + \frac{z }{n} \right ) ^ {n} ,$$

and has the following properties:

$$ e ^ {z _ {1} } e ^ {z _ {2} } = \ e ^ {z _ {1} + z _ {2} } \ \ \textrm{ and } \ \ ( e ^ {z _ {1} } ) ^ {z _ {2} } = \ e ^ {z _ {1} z _ {2} }$$

for any values of $ z _ {1} $and $ z _ {2} $.

For real $ x $, the graph of $ y = e ^ {x} $ (the exponential curve) passes through the point $ ( 0, 1) $and tends asymptotically to the $ x $-axis (see Fig.).

Figure: e036910a

In mathematical analysis one considers the exponential function $ y = a ^ {x} $for real $ x $and $ a > 0 $, $ a \neq 1 $; this function is related to the (basic) exponential function $ y = e ^ {x} $by

$$ a ^ {x} = e ^ {x \mathop{\rm ln} a } .$$

The exponential function $ y = a ^ {x} $is defined for all $ x $and is positive, monotone (it increases if $ a > 1 $and decreases if $ 0 < a < 1 $), continuous, and infinitely differentiable; moreover,

$$ ( a ^ {x} ) ^ \prime = a ^ {x} \mathop{\rm ln} a ,\ \ \int\limits a ^ {x} dx = \frac{a ^ {x} }{ \mathop{\rm ln} a } + C,$$

and in particular

$$ ( e ^ {x} ) ^ \prime = e ^ {x} ,\ \ \int\limits e ^ {x} dx = e ^ {x} + C ,$$

and in a neighbourhood of each point the exponential function can be expanded in a power series, for example:

$$ \tag{2 }e ^ {x} = \ 1 + \frac{x}{1!} + \dots + \frac{x ^ {n} }{n!} + \dots \equiv \ \sum _ { n= 0} ^ \infty \frac{x ^ {n} }{n!} .$$

The graph of $ y = a ^ {x} $is symmetric about the ordinate axis to the graph of $ y = ( 1/a) ^ {x} $. If $ a > 1 $, $ a ^ {x} $increases more rapidly than any power of $ x $as $ x \rightarrow + \infty $, while as $ x \rightarrow - \infty $it tends to zero more rapidly than any power of $ 1/x $, i.e. for any natural number $ b > 0 $,

$$ \lim\limits _ {x \rightarrow + \infty } \frac{a ^ {x} }{| x | ^ {b} } = \infty ,\ \ \lim\limits _ {x \rightarrow - \infty } | x | ^ {b} a ^ {x} = 0.$$

The inverse of an exponential function is a logarithmic function.

If $ a $and $ z $are complex, the exponential function is related to the (basic) exponential function $ w = e ^ {z} $by

$$ a ^ {z} = e ^ {z \mathop{\rm Ln} a } ,$$

where $ \mathop{\rm Ln} a $is the logarithm of the complex number $ a $.

The exponential function $ w = e ^ {z} $is a transcendental function and is the analytic continuation of $ y = e ^ {x} $from the real axis into the complex plane.

An exponential function can be defined not only by (1) but also by means of the series (2), which converges throughout the complex plane, or by Euler's formula

$$ e ^ {z} = e ^ {x+ iy } = e ^ {x} ( \cos y + i \sin y ).$$

If $ z = x + iy $, then

$$ | e ^ {z} | = e ^ {x} ,\ \ \mathop{\rm Arg} e ^ {z} = y + 2 \pi k,\ \ k = 0, \pm 1, \pm 2 , . . . .$$

The function $ e ^ {z} $is periodic with period $ 2 \pi i $: $ e ^ {z + 2 \pi i } = e ^ {z} $. The function $ e ^ {z} $assumes all complex values except zero; the equation $ e ^ {z} = a $has an infinite number of solutions for any complex number $ a \neq 0 $. These solutions are given by

$$ z = \mathop{\rm Ln} a = \mathop{\rm ln} | a | + i \mathop{\rm Arg} a.$$

The function $ e ^ {z} $is one of the basic elementary functions. It is used to express, for example, the trigonometric and hyperbolic functions.

Comments

For Euler's formula see also Euler formulas.

The basic exponential function $ z \mapsto \mathop{\rm exp} ( z) $defined by (1) or, equivalently, (2) (with $ z $instead of $ x $) is single-valued. However, powers $ z \mapsto a ^ {z} $for $ a $complex $ ( a \neq 0) $are multiple-valued since $ z \mapsto \mathop{\rm Ln} z $denotes the "multiple-valued inverse" to $ z \mapsto \mathop{\rm exp} ( z) $. Thus, since it is customary to abbreviate $ \mathop{\rm exp} ( z) $as $ e ^ {z} $, the left-hand side of the identity

$$ ( e ^ {z _ {1} } ) ^ {z _ {2} } = \ e ^ {z _ {1} z _ {2} }$$

is multiple-valued, while the right-hand side is single-valued. This identity is a dangerous one and should be dealt with with care, otherwise it may lead to nonsense like

$$ 1 = 1 ^ {1/2} = \ ( e ^ {2 \pi i } ) ^ {1/2} = \ e ^ {\pi i } = - 1.$$

By considering a single-valued branch of the logarithm (cf. Branch of an analytic function), or by considering the complete analytic function $ \mathop{\rm Ln} $on its associated Riemann surface, an awkward notation and a lot of confusion may disappear. For fixed $ a \in \mathbf C \setminus 0 $, any value (i.e. determination) of $ \mathop{\rm Ln} a $defines an exponential function:

$$ a ^ {z} = \ e ^ {z ( \textrm{ value } \textrm{ of } \mathop{\rm Ln} a) } .$$

References