What is an exponential

  • #1
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Definition/Summary

The exponential (the exponential function), written either [itex]e^x[/itex] 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([itex]i\pi[/itex]) = -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:

[tex]\frac{de^x}{dx}\ =\ e^x\ \text{and}\ e^0\,=\,1[/tex]

[tex]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!}[/tex]

[tex]e^x\ =\ \lim_{n\rightarrow\infty}\left(1\ +\ \frac{x}{n}\right)^n[/tex]

Euler's formula:

[tex]e^{ix}\ =\ cosx\ +\ i sinx[/tex]

and so cos and sin may be defined:

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

Hyperbolic functions:

[tex]e^{x}\ =\ coshx\ +\ sinhx[/tex]

[tex]coshx\ =\ \frac{1}{2}\left(e^{x}\ +\ e^{-x}\right)[/tex] and [tex]sinhx\ =\ \frac{1}{2}\left(e^{x}\ -\ e^{-x}\right)[/tex]

[tex]tanhx\ =\ \frac{sinhx}{coshx}\ =\ \frac{e^x\ -\ e^{-x}}{e^x\ +\ e^{-x}}[/tex]

[tex]tanh\frac{1}{2}x\ =\ \frac{e^x\ -\ 1}{e^x\ +\ 1}[/tex] and [tex]e^x\ =\ \frac{1\ +\ tanh\frac{1}{2}x}{1\ -\ tanh\frac{1}{2}x}[/tex]

Logarithms:

[tex]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[/tex]

[tex]e^{ln(x)}\ =\ x[/tex]

[tex]a^x\ =\ \left(e^{ln(a)}\right)^x\ =\ e^{x\,ln(a)}[/tex]

[tex]y\ =\ a^x \Leftrightarrow\ x\ =\ log_a(y)\ \equiv\ \frac{ln(y)}{ln(a)}[/tex]


[tex]\frac{da^x}{dx}\ =\ ln(a)\,e^{x\,ln(a)}\ =\ ln(a)\,a^x[/tex]

Extended explanation

"Exponentially" ("geometrically"):

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

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

(For example, 2x 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!
 
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Answers and Replies

  • #2
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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).
 

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