Difference between revisions of "Exponential functions"
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| − | { | + | 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 | $\Delta$-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 | $\Delta$-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 | |
| − | + | $$y^{\nabla}=y;y(s)=1,$$ | |
| − | |$\ | + | defining the [[nabla exponential | $\nabla$-exponential]] functions. |
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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 | |
| − | + | $$Dy=y; y(s)=1.$$ | |
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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 $$Dy=y; y(s)=1.$$
