A Laplace Transform

[Prometheus]

So I want to find the Laplace transform of:

 

[latex] f(t)=\alpha e^{-\alpha t} + \beta e^{-\beta t}[/latex]

 

I make it:

 

[latex]f^*(s)=\frac{\alpha}{\alpha+s}+\frac{\beta}{\beta+s}[/latex]

 

Which I thought simply follows from the linearity of the integral operator.

 

But according to my lecture notes it is:

 

[latex] f^*(s)=(\frac{\alpha}{\alpha+s})(\frac{\beta}{\beta+s}) [/latex]

 

I'm hoping there is a mistake in the lecture notes, otherwise I've totally misunderstood things. I would appreciate it if anyone could comment on which answer is correct.

 

 

[imatfaal]

http://www.wolframalpha.com/input/?i=laplace+transform+f%28t%29%3Dxe^{-yt}+%2B+ye^{-yt}

[Prometheus]

Cheers. Good old wolfram alpha, didn't even occur to me to check there.

[Prometheus]

Turns out it was the product, not the sum. What I neglected to add was that I was dealing with random variables. I knew this was true for Gaussian random variables, but didn't know it extends to all random variables (which what it seems to be true, i'll have to find out exactly why, but I know it's related to convolution theorem).