Convolución Integrales – Tabla

Referencia: Lathi Tabla 2.1 p176

Respuesta a estado cero y convolución con Sympy-Python

No	x1(t)	x2(t)	x1(t)⊗x2(t) = x2(t)⊗x1(t)
1	x(t)	δ(t-T)	x(t-T)
2	eλt μ(t)	μ(t)	$\frac{1-e^{\lambda t}}{-\lambda} \mu (t)$
3	μ(t)	μ(t)	t μ(t)
4	eλ1t μ(t)	eλ2t μ(t)	$\frac{e^{\lambda _1 t}-e^{\lambda _2 t}}{\lambda _1 - \lambda _2} \mu(t)$
			$\lambda _1 \neq \lambda _2$
5	eλt μ(t)	eλt μ(t)	t eλt μ(t)
6	t eλt μ(t)	eλt μ(t)	$\frac{1}{2} t^2 e^{\lambda t} \mu(t)$
7	t N μ(t)	eλt μ(t)	$\frac{N! e^{\lambda t}}{\lambda^{N+1}} \mu(t) - \sum_{k=0}^{N}\frac{N! t^{N-k}}{\lambda^{N+1} (N-k)!} \mu(t)$
8	t M μ(t)	t N μ(t)	$\frac{M! N!}{(M+N+1)!} t^{M+N+1} \mu (t)$
9	t eλ1t μ(t)	eλ2t μ(t)	$\frac{e^{\lambda_2 t}- e^{\lambda_1 t} + (\lambda_1-\lambda_2)te^{\lambda_1 t}}{(\lambda_1-\lambda_2)^2} \mu (t)$
10	t M eλt μ(t)	t N eλt μ(t)	$\frac{M! N!}{(M+N+1)!} t^{M+N+1} e^{\lambda t}\mu (t)$
11	t M eλ1t μ(t)	$t^N e^{\lambda_2t} \mu (t)$	$\sum_{k=0}^{M}\frac{(-1)^k M! (N+k)! t^{M-k} e^{\lambda_1t}}{k!(M-k)!(\lambda_1 - \lambda_2)^{N+k+1}} \mu (t)$
	λ1 ≠λ2		$+ \sum_{k=0}^{N}\frac{(-1)^k N! (M+k)! t^{N-k} e^{\lambda_2t}}{k!(N-k)!(\lambda_2 - \lambda_1)^{M+k+1}} \mu (t)$
12	$e^{\alpha t} \cos (\beta t + \theta) \mu (t)$	eλt μ(t)	$\frac{\cos(\theta - \phi)e^{\lambda t}-e^{-\alpha t} \cos(\beta t + \theta - \phi)}{\sqrt{(\alpha + \lambda)^2 + \beta^2}} \mu (t)$
			$\phi =tan^{-1} \Big[\frac{-\beta}{(\alpha + \lambda})\Big]$
13	eλ1t μ(t)	eλ2t μ(-t)	$\frac{e^{\lambda_1 t} \mu (t) + e^{\lambda_2 t} \mu (-t)}{\lambda_2 -\lambda_1}$
			$\text{Re} \lambda_2 > \text{Re} \lambda_1$
14	eλ1t μ(-t)	eλ2t μ(-t)	$\frac{e^{\lambda_1 t} -e^{\lambda_2 t}}{\lambda_2 -\lambda_1} \mu (-t)$