I have read questions and answers about this topic and i am still confused, using this formula we can calculate the Laplace transform of a product of two functions:
$$ L[a_{(t)} b_{(t)}]={{1}\over{2 i \pi}} \int_{\sigma -i \infty}^{\sigma +i \infty}A_{(z)}B_{(s-z)}dz $$
But when i test this formula on an example i get wrong result. My example is: a(t)=t , b(t)=e^(-t)
So the correct answer should be
$$ L[te^{-t}]={{1}\over{(s+1)^2}} $$
But when i substitute:
$$ L[t]={{1}\over{s^2}} $$
$$ L[e^{-t}]={{1}\over{s+1}} $$
into the above formula, i get:
$$ L[te^{-t}]={{1}\over{2 \pi i}}\int_{\sigma-i \infty}^{\sigma + i \infty} {{1}\over{z^2}}{{1}\over{s-z+1}}dz=\sum res $$
Residue at z=0 is 1/(s+1)^2 and residue at z=s+1 is also 1/(s+1)^2 so the result i get using this formula is twice the correct result.
Where did i go wrong? Is there an article i can read about this formula?