Difference between revisions of "Exponential"

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(Created page with "Let $a$ be constant. Exponential functions are of the form $$f(x)=a^x.$$ The most commonly used exponential function is $e^x$, where $e$ is the E | base of the natural logar...")
 
 
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Let $a$ be constant. Exponential functions are of the form
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__NOTOC__
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$$f(x)=a^x.$$
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The most commonly used exponential function is $e^x$, where $e$ is the [[E | base of the natural logarithm]]. It can be characterized as the unique solution to the initial value problem
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The exponential function $\exp \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined by the formula
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$$\left\{ \begin{array}{ll}
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$$\exp(z) \equiv e^z = \sum_{k=0}^{\infty} \dfrac{z^k}{k!},$$
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y'=y \\
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where $e$ is the [[E | base of the natural logarithm]].
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y(0)=1.
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\end{array} \right.$$
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<div align="center">
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<gallery>
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File:Exponentialplot.png|Graph of $\exp$.
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File:Complexexponentialplot.png|[[Domain coloring]] of $\exp$.
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</gallery>
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</div>
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=Properties=
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[[Derivative of the exponential function]]<br />
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[[Taylor series of the exponential function]]<br />
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[[Euler E generating function]]<br />
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[[Continued fraction for 1/sqrt(pi) integral from -infinity to infinity of e^(-t^2)/(z-t) dt]]<br />
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[[Exponential function is periodic with period 2pii]]<br />
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[[Euler's formula]]<br />
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[[E^(-x/(1-x)) is less than 1-x is less than e^(-x) for nonzero real x less than 1]]<br />
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[[E^x is greater than 1+x for nonzero real x]]<br />
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[[E^x is less than 1/(1-x) for nonzero real x less than 1]]<br />
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[[X/(1+x) less than 1-e^(-x) less than x for nonzero real x greater than -1]]<br />
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[[X less than e^x-1 less than x/(1-x) for nonzero real x less than 1]]<br />
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[[1+x greater than exp(x/(1+x)) for nonzero real x greater than -1]]<br />
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[[E^x greater than 1+x^n/n! for n greater than 0 and nonzero real x greater than 0]]<br />
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[[E^x greater than (1+x/y)^y greater than exp(xy/(x+y) for x greater than 0 and y greater than 0)]]<br />
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[[E^(-x) less than 1-(x/2) for 0 less than x less than or equal to 1.5936]]<br />
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[[Abs(z)/4 less than abs(e^z-1) less than (7abs(z))/4 for 0 less than abs(z) less than 1]]<br />
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[[Abs(e^z-1) less than or equal to e^(abs(z))-1 less than or equal to abs(z)e^(abs(z))]]<br />
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[[Sum of cosh and sinh]]<br />
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[[Difference of cosh and sinh]]<br />
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==References==
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* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=findme|next=Logarithm (multivalued) of the exponential}}: $4.2.1$
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[[Category:SpecialFunction]]
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[[Category:Definition]]

Latest revision as of 23:37, 21 October 2017


The exponential function $\exp \colon \mathbb{C} \rightarrow \mathbb{C}$ is defined by the formula $$\exp(z) \equiv e^z = \sum_{k=0}^{\infty} \dfrac{z^k}{k!},$$ where $e$ is the base of the natural logarithm.

Properties

Derivative of the exponential function
Taylor series of the exponential function
Euler E generating function
Continued fraction for 1/sqrt(pi) integral from -infinity to infinity of e^(-t^2)/(z-t) dt
Exponential function is periodic with period 2pii
Euler's formula
E^(-x/(1-x)) is less than 1-x is less than e^(-x) for nonzero real x less than 1
E^x is greater than 1+x for nonzero real x
E^x is less than 1/(1-x) for nonzero real x less than 1
X/(1+x) less than 1-e^(-x) less than x for nonzero real x greater than -1
X less than e^x-1 less than x/(1-x) for nonzero real x less than 1
1+x greater than exp(x/(1+x)) for nonzero real x greater than -1
E^x greater than 1+x^n/n! for n greater than 0 and nonzero real x greater than 0
E^x greater than (1+x/y)^y greater than exp(xy/(x+y) for x greater than 0 and y greater than 0)
E^(-x) less than 1-(x/2) for 0 less than x less than or equal to 1.5936
Abs(z)/4 less than abs(e^z-1) less than (7abs(z))/4 for 0 less than abs(z) less than 1
Abs(e^z-1) less than or equal to e^(abs(z))-1 less than or equal to abs(z)e^(abs(z))
Sum of cosh and sinh
Difference of cosh and sinh

References

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