Difference between revisions of "Exponential functions"

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(Created page with "{| class="wikitable" |+Time Scale Exponential Functions |- |$\mathbb{T}=$ |$e_p(t,s)=$ |- |$\mathbb{R}$ |$\begin{array}{ll} e_p(t,s) &= \exp \left( \displaystyle\int_s^t \disp...")
 
 
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{| class="wikitable"
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The classical exponential function $e^{x-s}$ is the unique solution to the initial value problem
|+Time Scale Exponential Functions
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$$y'=y; y(s)=1.$$
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The standard way to generalize this to time scales is called the [[Delta exponential | $\Delta$-exponential]] function, which is the solution of
|$\mathbb{T}=$
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$$y^{\Delta}=y;y(s)=1.$$
|$e_p(t,s)=$
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It generalizes the above equation in the sense that the classical derivative is replaced by the [[Delta derivative | $\Delta$-derivative]] on some time scale. If instead of using the $\Delta$-derivative one uses the [[nabla derivative | $\nabla$-derivative]] then the resulting exponential equation is
|-
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$$y^{\nabla}=y;y(s)=1,$$
|$\mathbb{R}$
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defining the [[nabla exponential | $\nabla$-exponential]] functions.
|$\begin{array}{ll}
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e_p(t,s) &= \exp \left( \displaystyle\int_s^t \displaystyle\lim_{h \rightarrow 0} \dfrac{1}{h} \log(1 + hp(\tau)) d\tau \right) \\
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Generally speaking, given some kind of time scale derivative operator $D$, we can define exponential functions by the $D$-dynamic equation
&\hspace{-10pt} \stackrel{\mathrm{L'Hôpital}}{=} \exp \left( \displaystyle\int_s^t \displaystyle\lim_{h \rightarrow 0} \dfrac{1}{1+hp(\tau)} p(\tau) d\tau \right) \\
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$$Dy=y; y(s)=1.$$
&= \exp \left( \displaystyle\int_s^t p(\tau) d \tau \right)
 
\end{array}$
 
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|Butter
 
|Ice cream
 
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Latest revision as of 20:55, 20 October 2014

The classical exponential function $e^{x-s}$ is the unique solution to the initial value problem $$y'=y; y(s)=1.$$ The standard way to generalize this to time scales is called the $\Delta$-exponential function, which is the solution of $$y^{\Delta}=y;y(s)=1.$$ It generalizes the above equation in the sense that the classical derivative is replaced by the $\Delta$-derivative on some time scale. If instead of using the $\Delta$-derivative one uses the $\nabla$-derivative then the resulting exponential equation is $$y^{\nabla}=y;y(s)=1,$$ defining the $\nabla$-exponential functions.

Generally speaking, given some kind of time scale derivative operator $D$, we can define exponential functions by the $D$-dynamic equation $$Dy=y; y(s)=1.$$