Difference between revisions of "Covolution theorem for unilateral Laplace transform"

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==Theorem==
 
==Theorem==
 
The following formula holds:
 
The following formula holds:
$$\widehat{(f*g)}(t,s)=\displaystyle\int_s^t \hat{f}(t,\sigma(\xi))\hat{g}(\xi,s) \Delta \xi,$$
+
$$\mathscr{L}_{\mathbb{T}}\{f*g\}(z)=\mathscr{L}\{f\}(z) \mathscr{L}\{g\}(z),$$
where $\widehat{(f*g)}$ denotes the solution of the [[shifting problem]] and $(f*g)$ denotes the [[convolution]].
+
where $\mathscr{L}$ denotes the [[unilateral Laplace transform]] and $f*g$ denotes the [[unilateral convolution]].
  
 
==Proof==
 
==Proof==

Latest revision as of 13:44, 20 January 2023

Theorem

The following formula holds: $$\mathscr{L}_{\mathbb{T}}\{f*g\}(z)=\mathscr{L}\{f\}(z) \mathscr{L}\{g\}(z),$$ where $\mathscr{L}$ denotes the unilateral Laplace transform and $f*g$ denotes the unilateral convolution.

Proof

References