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exponential
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Definition/Summary
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The exponential (the exponential function), written either  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( ) = -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). |
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Equations
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Definitions:
Euler's formula:
and so cos and sin may be defined:
 and 
Hyperbolic functions:
 and 
 and 
Logarithms:
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Recent forum threads on exponential
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Breakdown
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Mathematics
> Calculus/Analysis
>> Functions
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Extended explanation
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"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) |
Commentary
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. Thankyou jwhipple for pointing out these mistakes 
I've completely rewritten it.
