I'm trying to find the Laplace Transform of a Convolution (in the form of an integral), but am having trouble discovering one of the two convoluted functions, f(t) and g(t).

Question

mendicant_bias · Answer

The original question is:

$$\mathcal{L}\left\{\vphantom{}\int\limits_{0}^{t}e^{\tau}d \tau\right\};$$
What I'm confused about is that I can immediately see that one function (let's say f(t)) is e^t; I'm not sure what I can or should assign g(t) being, because I don't see anything in the integrand of the form g(t-tau), and I don't see anything outside the integrand multiplying the whole thing by t. Any help would be appreciated.

mendicant_bias · Answer

$$f(t)=e^{t}, \ g(t)= \ ?$$

mendicant_bias · Answer

Does $$g(t)=1$$since whatever I select for g(t) seems arbitrary and unrelated to either t or tau, but cannot multiply the function by some value it already isn't multiplied by?

(e.g., if I had some function):$$\int\limits_{0}^{t}5 e^{\tau}d \tau,$$is it reasonable for me to say that $$f(t)=e^{t}, \ f(\tau)=e^{\tau}; \ \ \ g(t)=5, \ g(t-\tau)=5$$

mendicant_bias · Answer

@midhun.madhu1987

mendicant_bias · Answer

@Zarkon , are you any good with convolution?

mendicant_bias · Answer

@wio , could you maybe help me on this? Any help would be much appreciated.

anonymous · Answer

$$g(t-\tau)=1$$

mendicant_bias · Answer

Alright, so then g(t) would just be one as well, right?

If you have time to help me with another problem, I posted it and can tag you in it.