I've come into contact with this two part question, and the latter I'm not too sure how to go about; at least to me upon researching, I can't find anything remotely similar to what I've been asked.
The question is as follows:
a) Show that $$\mathcal L\left[\int_{0}^t f(\tau)d\tau\right] = \frac1p\mathcal L[f(t)]$$
this I had no problem with, and was able to prove easily. The following part is what I'm struggling with.
b) Hence, calculate the LT of the so-called Fresnel integrals defined as follows:
$$C(t) = \int_{0}^t \frac{\cos(\tau)}{\sqrt{2\pi\tau}}d\tau $$ and
$$S(t) = \int_{0}^t \frac{\sin(\tau)}{\sqrt{2\pi\tau}}d\tau $$
I assumed to use the above identity (given the choice of hence in the question), and applying this, have ended up with:
$$ \frac1{p\sqrt{2\pi}}\mathcal L\big(\frac{\cos{t}}{\sqrt{t}}\big)$$
and $$ \frac1{p\sqrt{2\pi}}\mathcal L\big(\frac{\sin{t}}{\sqrt{t}}\big)$$
From here, I've hit a brick wall; I've tried looking in various tables, and other LT methods, but I can't seem to find one that will do the trick here. I'm not too sure how to deal with the remaining Laplace transforms I'm left with.
Any small hints / tips to put me back on track would be much appreciated!!
EDIT
After using Robert's and jmerry's hints, I now have:
$$\mathcal L \left[C(t)\right] = \frac{\sqrt{2}}{4p} \left( \frac1{\sqrt{p-i}}+\frac1{\sqrt{p+i}}\right) $$
$$\mathcal L \left[S(t)\right] = \frac{-i\sqrt{2}}{4p} \left( \frac1{\sqrt{p-i}}-\frac1{\sqrt{p+i}}\right) $$
as my result for C(t) and S(t).
Please verify for me 🙂
Best Answer
Hint: $\cos(\tau)$ is the real part of $\exp(i\tau)$, which you combine with the $\exp(-p \tau)$ in one integral, and use the definition of the Gamma function.