Laplace transform restriction and differentiation

Question

every one.I have just started learning Laplace transform.However, there are two main conceptual problems I can't convince myself.

The first problem is about the restriction of this integral, I understand that for a real variable s, it needs to be positive so that the integral converges to a value, e.g. f(t)=1 then L(f(t)) = 1/s . s>0 But,if the s is a complex variable, I don't understand why the book says , e.g. for f(t) = exp(-at) ,then Re(s+a)>0.

Why did they simply ignore the imaginary part of variable s, or are they just considering the real part of the function?(which they didn't mention)?

The second question is when I was asked to verify L(tsinat) =3 http://tutorial.math.lamar.edu/Classes/DE/Laplace_Table_files/eq0021M.gif by using L(cosat) =

the problem is why do we have to differentiate this formula with respect to a? I thought a was a constant value which can't be differentiated in many problems I encountered while solving ODEs before.

Thank you so much for helping! Greatly appreciated.

Answer 1

For the first question: If $s=\sigma + i\omega$ and $t$ is real, then $$|e^{-st}|=|e^{-\sigma t}e^{-i\omega t}| = e^{-\sigma t}$$ since $|e^{i\omega t}|=1$. In other words it's only the real part of $s$ that determines where the ontegral converges.

Answer 2

Hint for the second part: Use this fact that:

$$\mathcal{L}(t^nf(t))=(-1)^n\frac{d^n F(s)}{ds^n}$$ wherein $\mathcal{L}(f(t))=F(s).$