A Literally Complex Integral

Note that $\dfrac d{dx} e^{\frac {\pi i}{2x}} = - \frac {\pi i}{2x^2} e^{\frac {\pi i}{2x}}$ . Therefore,

$\begin{aligned} I & = i^3 \int_a^\infty \frac {e^{\frac {\pi i}{2x}}}{x^2} dx \\ & = i^3 \cdot \frac 2{-\pi i} {\color{#3D99F6}e^{\frac {\pi i}{2x}}} \ \bigg|_a^\infty & \small \color{#3D99F6} \text{By Euler's formula: }e^{\theta i} = \cos \theta + i \sin \theta \\ & = \frac 2\pi \left[\cos \frac \pi{2x} + i \sin \frac \pi{2x} \right]_a^\infty \\ & = \frac 2\pi \left[\cos 0 + i \sin 0 - \cos \frac \pi{2a} - i \sin \frac \pi{2a}\right] \\ & = \frac 2\pi \left[1 - \cos \frac \pi{2a} - i \sin \frac \pi{2a}\right] \end{aligned}$

For $I = \dfrac 4\pi \implies \cos \dfrac \pi {2a} = -1$ and $\sin \dfrac \pi{2a} = 0$ , $\implies a = \pm \dfrac 12$ therefore, the largest $a= \dfrac 12 = \boxed{0.5}$ .