# Difference between revisions of "Relationship between nabla exponential and delta exponential"

From timescalewiki

Line 1: | Line 1: | ||

− | + | ==Theorem== | |

− | + | If $p$ is [[continuous]] and $\nu$-regressive then | |

$$\hat{e}_p(t,s)=e_{\frac{p^{\sigma}}{1-p^{\sigma}\nu}}(t,s)=e_{\ominus(-p^{\sigma})}(t,s),$$ | $$\hat{e}_p(t,s)=e_{\frac{p^{\sigma}}{1-p^{\sigma}\nu}}(t,s)=e_{\ominus(-p^{\sigma})}(t,s),$$ | ||

where $\hat{e}_p$ denotes the [[nabla exponential|$\nabla$-exponential]] and $e_p$ denotes the [[Delta exponential|$\Delta$-exponential]]. | where $\hat{e}_p$ denotes the [[nabla exponential|$\nabla$-exponential]] and $e_p$ denotes the [[Delta exponential|$\Delta$-exponential]]. | ||

− | + | ||

− | + | ==Proof== | |

− | + | ||

− | + | ==References== | |

+ | |||

+ | [[Category:Theorem]] | ||

+ | [[Category:Unproven]] |

## Latest revision as of 22:22, 9 June 2016

## Theorem

If $p$ is continuous and $\nu$-regressive then $$\hat{e}_p(t,s)=e_{\frac{p^{\sigma}}{1-p^{\sigma}\nu}}(t,s)=e_{\ominus(-p^{\sigma})}(t,s),$$ where $\hat{e}_p$ denotes the $\nabla$-exponential and $e_p$ denotes the $\Delta$-exponential.