Showing an inequality

Question

I wish to show $$|{(Re^{i \theta})^{-\frac{1}{2}}}\exp(\frac{-1}{Re^{i \theta}})| < \frac{M}{R^k}$$ for some M, k > 0

I've managed to reduce it to $$|R^{-\frac{1}{2}}| |\exp(\frac{-1}{Re^{i \theta}})|$$ but am unsure of where to go from here.

In context, I'm trying to find the inverse Laplace transform of $s^{\frac{-1}{2}}\exp(\frac{-1}{s})$ using the complex inversion formula. Any help at all is greatly appreciated!

Answer 1

Hints :

If $a$ and $b$ are two real numbers,

$$\left|\exp(a+ib)\right|=\exp(a)$$

If $\theta$ is a real number, $R>0$ :

$$-\dfrac 1{Re^{i\theta}}=-\dfrac 1R e^{-i\theta}=-\dfrac 1R \cos(\theta)+\dfrac iR\sin(\theta)$$