So i’m applying the inverse Laplace Transform to the following function:

$$

frac{1}{s} exp(-sqrt{s/k}x)

$$

Thus, the inverse Laplace transform is given by:

$$

u(x,t) = frac{1}{2pi i}int^{gamma+iinfty}_{gamma-iinfty} frac{1}{s}expbigg(-sqrt{frac{s}{k}}xbigg) exp(st) ds.

$$

the following contour is used as we have a multivalued function:

(enter image description here)(1)

For the line AB. we parameterise by letting $s = u exp (i pi)$

For the line ED, we parameterise by letting $s = u exp (-i pi)$

For the small circle C, we parameterise by letting $s = epsilon exp(i theta)$

we can assume that $K$ and $F$ go to zero as $R rightarrow infty$.

**My question is:**

I want to prove that the arcs given by $K$ and $F$ go to zero as $R rightarrow infty$.

How do you parameterise the arcs $K$ and $F$ so that they go to zero as $R rightarrow infty$?

I know that you are supposed to use the ML inequality/estimation lemma but these arcs do not look like a semicircular arc or a quarter of a circle so I’m stuck here.

(1): https://i.stack.imgur.com/TCyaJ.png