Exponential Definition & Summary: An Overview

[Greg Bernhardt]

Definition/Summary

The exponential (the exponential function), written either $e^x$ or exp(x), is the only function whose derivative (apart from a constant factor) is itself.

It may be defined over the real numbers, over the complex numbers, or over more complicated algebras such as matrices.

Its value at 0 is 1, and its value at 1 is the exponential constant (or Euler's constant or Napier's constant), e = 2.71828...

Its value at pure imaginary numbers is a combination of cos and sin (and therefore it may be used to define them): exp(ix) = cosx + isinx (Euler's formula), and therefore exp($i\pi$) = -1 (Euler's indentity).

Its inverse (over real or complex numbers) is the natural logarithm, log(x) (often written ln(x), to distinguish it from the base-10 logarithm): if y = exp(x), then x = log(y).

Equations

Definitions:

$$\frac{de^x}{dx}\ =\ e^x\ \text{and}\ e^0\,=\,1$$

$$e^x\ =\ 1\ +\ x\ +\ \frac{x^2}{2} +\ \frac{x^3}{6} +\ \frac{x^4}{24} +\ \frac{x^5}{120}\ \dots\ = \sum_{n\,=\,0}^{\infty}\frac{x^n}{n!}$$

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

Euler's formula:

$$e^{ix}\ =\ cosx\ +\ i sinx$$

and so cos and sin may be defined:

$$cosx\ =\ \frac{1}{2}\left(e^{ix}\ +\ e^{-ix}\right)$$ and $$i sinx\ =\ \frac{1}{2}\left(e^{ix}\ -\ e^{-ix}\right)$$

Hyperbolic functions:

$$e^{x}\ =\ coshx\ +\ sinhx$$

$$coshx\ =\ \frac{1}{2}\left(e^{x}\ +\ e^{-x}\right)$$ and $$sinhx\ =\ \frac{1}{2}\left(e^{x}\ -\ e^{-x}\right)$$

$$tanhx\ =\ \frac{sinhx}{coshx}\ =\ \frac{e^x\ -\ e^{-x}}{e^x\ +\ e^{-x}}$$

$$tanh\frac{1}{2}x\ =\ \frac{e^x\ -\ 1}{e^x\ +\ 1}$$ and $$e^x\ =\ \frac{1\ +\ tanh\frac{1}{2}x}{1\ -\ tanh\frac{1}{2}x}$$

Logarithms:

$$y\ =\ e^x \Leftrightarrow\ x\ =\ ln(y) \Leftrightarrow\ \frac{dy}{dx}\ =\ y\ \text{and}\ y(0)\,=\,1\Leftrightarrow\ \frac{dx}{dy}\ =\ \frac{1}{x}\ \text{and}\ x(1)\,=\,0$$

$$e^{ln(x)}\ =\ x$$

$$a^x\ =\ \left(e^{ln(a)}\right)^x\ =\ e^{x\,ln(a)}$$

$$y\ =\ a^x \Leftrightarrow\ x\ =\ log_a(y)\ \equiv\ \frac{ln(y)}{ln(a)}$$$$\frac{da^x}{dx}\ =\ ln(a)\,e^{x\,ln(a)}\ =\ ln(a)\,a^x$$

Extended explanation

"Exponentially" ("geometrically"):

A function is said to increase exponentially (or geometrically), or is O(e^x), if it increases "as fast as" e^x

So such a function increases faster than any fixed power of x.

(For example, 2^x increases exponentially.

By comparison, a function increases arithmetically, or is O(x), if it increases "as fast as" x, and is O(xⁿ) if it increases "as fast as" xⁿ)

* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!

 

[fresh_42]

The exponential function is probably the most universal. It appears everywhere in nature, and so in physics and mathematics. It is our template for integration. Differentiation is a linear approximation of something curved. It translates multiplication into addition:
$$
\left. \dfrac{d}{dx}\right|_{x=p}\left( f(x)\cdot g(x)\right) =\left(\left. \dfrac{d}{dx}\right|_{x=p} f(x)\right)\cdot g(x)+f(x)\cdot\left(\left. \dfrac{d}{dx}\right|_{x=p} g(x)\right)
$$
and the exponential function reverses this: $\exp(a) +\exp(b)=\exp(a\cdot b)$. The most beautiful way to see this is in my opinion the formula (eq. 61 in https://www.physicsforums.com/insights/pantheon-derivatives-part-iv/)
$$
\exp \circ \operatorname{ad} = \operatorname{Ad} \circ \exp
$$
which connects the adjoint representation of a Lie group with the adjoint representation of its Lie algebra (tangent space of the group).