- #1

Telemachus

- 835

- 30

[tex]f(t)= e^{-t} H(t)[/tex]

[tex]g(t)=e^t H(t)[/tex] H(t) the unit step Heaviside function.

So I have to find: [tex]f \star g[/tex]

This is what I did:

[tex]f \star g=\displaystyle\int_{-\infty}^{\infty}e^{-\lambda}e^{t-\lambda}H(\lambda)H(t-\lambda)d\lambda=\displaystyle\int_{-\infty}^{\infty}e^{t-2\lambda} \left [ H(\lambda)-H(\lambda-t) \right ] d\lambda=[/tex]

[tex]=\displaystyle\int_{-\infty}^{\infty}e^{t-2\lambda}H(\lambda)d\lambda-\displaystyle\int_{-\infty}^{\infty}e^{t-2\lambda} H(\lambda-t)d\lambda=\displaystyle\int_{0}^{\infty}e^{t-2\lambda}d\lambda-\displaystyle\int_{t}^{\infty}e^{t-2\lambda} d\lambda=\displaystyle\int_{0}^{t}e^{t-2\lambda} d\lambda=-2e^{-t}+2e^{t}[/tex]

Is this fine?