Difference between revisions of "Delta Taylor's formula"

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Let $\mathbb{T}$ be a [[time scale]].  
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==Theorem==
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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
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$$f(t)=\displaystyle\sum_{k=0}^{n-1} h_k(t,\alpha) f^{\Delta^k}(\alpha) + \displaystyle\int_{\alpha}^{\rho^{n-1}(t)} h_{n-1}(t,\sigma(\tau)) f^{\Delta^n}(\tau) \Delta \tau,$$
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where $h_k$ denotes the [[Delta hk|$h_k$ Taylor monomials]].
  
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<strong>Theorem:</strong> Let $n \in \{1,2,\ldots\}$. Suppose $f$ is $n$-times differentiable on $\mathbb{T}^{\kappa^n}$. Let $\alpha \in \mathbb{T}^{\kappa^{n-1}}, t\in\mathbb{T}$ then
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==Proof==
$$f(t)=\displaystyle\sum_{k=0}^{n-1} h_k(t,\alpha) f^{\Delta^k}(\alpha) + \displaystyle\int_{\alpha}^{\rho^{n-1}(t)} h_{n-1}(t,\sigma(\tau)) f^{\Delta^n}(\tau) \Delta \tau,$$
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where $h_k$ denotes the [[Polynomials | $h_k$ polynomials]].
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==References==
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<strong>Proof:</strong> █
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[[Category:Theorem]]
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[[Category:Unproven]]
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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