Exponential of complex square root

Question

Is there any way to simplify any further the exponential of a complex square root, as in the following expression:

$$ e^{ a + \sqrt{x + i\cdot y}}, $$

where $a>0, x >0$ and $y<0$. If I were to select the principal square root, I could define $ r = \sqrt{x^2 + y^2}$ and $\theta = \arctan {x/y}$. Then,

$$ e^{ a + \sqrt{r}\cdot(\cos(\theta/2) + i\cdot \sin(\theta/2) )}. $$

Is there a way to get a friendlier or simplify? I have to later on integrate this expression with respect to $y$ and it doesn't seem easy to integrate.

Answer 1

The integration does not seem to be very difficult.

Let $$\sqrt{x+i y}=t \implies y=i \left(x-t^2\right)\implies dy=-2it\,dt$$ $$\int e^{a+\sqrt{x+i y}}\,dy=-2i e^a\int t e^{\sqrt{t^2}}\,dt$$ Simplify and use one integration by parts.