quantum mechanics - Landau & Lifshitz's Approach (contour method) on the WKB connection formulas

Background of the question (see pp. 161, section 47 in Landau & Lifshitz's quantum mechanics textbook Vol3, 2nd Ed. Pergamon Press). We a following potential well $$U(x)\leq E \quad\text{for} \quad x \leq a ,$$ $$U(x)>E \quad\text{for} \quad x>a .\tag{47.0}$$

The WKB solutions right and left to the turning point are

$$\psi=\dfrac{C}{2\sqrt{p}}\exp{\left(-\dfrac{1}{\hbar}\left|\int_a^x pdx\right|\right)} \quad \text{for} \quad x>a, \tag{47.1}$$

$$\psi=\dfrac{C_1}{\sqrt{p}}\exp{\left(\dfrac{i}{\hbar}\int_a^x pdx\right)}+\dfrac{C_2}{\sqrt{p}}\exp{\left(-\dfrac{i}{\hbar}\int_a^x p dx\right)}\quad \text{for} \quad x

respectively. Most quantum mechanics textbooks determine the relation between $C$ and $C_i$'s by find the exact solution near the turning point. And then let the exact solution match with the WKB solutions.

However, in Landau & Lifshitz's quantum mechanics textbooks (vol3, section 47) they let $x$ vary in the complex plane and pass around the turning point $a$ from right to left through a large semicircle in the upper complex plane. Landau claims starting from +$\infty$, when arrive at $-\infty$ (left to $a$), there is phase gain $\pi$ in the denominator of prefactor in Eq.(2). From this we can determine $$C_2=\frac{C}{2}\exp\left(i\frac{\pi}{4}\right).\tag{47.4a}$$

They also claim the first term will exponential decay along the semicircle in the upper half plane. Question is why? Can we show $$\Im{\left(\int_a^x pdx\right)},$$ where $\Im$ stands for the imaginary part, is positive?