Difference between revisions of "Delta Taylor's formula"

From timescalewiki
Jump to: navigation, search
 
(One intermediate revision by the same user not shown)
Line 2: Line 2:
 
Let $\mathbb{T}$ be a [[time scale]] and $n \in \{1,2,\ldots\}$. Suppose $f$ is $n$-times differentiable on $\mathbb{T}^{\kappa^n}$. Let $\alpha \in \mathbb{T}^{\kappa^{n-1}}, t\in\mathbb{T}$ then
 
Let $\mathbb{T}$ be a [[time scale]] and $n \in \{1,2,\ldots\}$. Suppose $f$ is $n$-times differentiable on $\mathbb{T}^{\kappa^n}$. Let $\alpha \in \mathbb{T}^{\kappa^{n-1}}, t\in\mathbb{T}$ then
 
$$f(t)=\displaystyle\sum_{k=0}^{n-1} h_k(t,\alpha) f^{\Delta^k}(\alpha) + \displaystyle\int_{\alpha}^{\rho^{n-1}(t)} h_{n-1}(t,\sigma(\tau)) f^{\Delta^n}(\tau) \Delta \tau,$$
 
$$f(t)=\displaystyle\sum_{k=0}^{n-1} h_k(t,\alpha) f^{\Delta^k}(\alpha) + \displaystyle\int_{\alpha}^{\rho^{n-1}(t)} h_{n-1}(t,\sigma(\tau)) f^{\Delta^n}(\tau) \Delta \tau,$$
where $h_k$ denotes the [[Polynomials | $h_k$ polynomials]].
+
where $h_k$ denotes the [[Delta hk|$h_k$ Taylor monomials]].
  
  
Line 8: Line 8:
  
 
==References==
 
==References==
*{{PaperReference|Analysis of the bilateral Laplace transform on time scales with applications|2021|Tom Cuchta|author2=Svetlin Georgiev|prev=|next=}}: Theorem 11
 
  
 
[[Category:Theorem]]
 
[[Category:Theorem]]
 
[[Category:Unproven]]
 
[[Category:Unproven]]

Latest revision as of 17:05, 15 January 2023

Theorem

Let $\mathbb{T}$ be a time scale and $n \in \{1,2,\ldots\}$. Suppose $f$ is $n$-times differentiable on $\mathbb{T}^{\kappa^n}$. Let $\alpha \in \mathbb{T}^{\kappa^{n-1}}, t\in\mathbb{T}$ then $$f(t)=\displaystyle\sum_{k=0}^{n-1} h_k(t,\alpha) f^{\Delta^k}(\alpha) + \displaystyle\int_{\alpha}^{\rho^{n-1}(t)} h_{n-1}(t,\sigma(\tau)) f^{\Delta^n}(\tau) \Delta \tau,$$ where $h_k$ denotes the $h_k$ Taylor monomials.


Proof

References