How to integrate Fresnel Integrals? $\int_0^y e^\frac{-j\beta(z)^2}\rho dz$

Question

I am having trouble solving this integration of a spherical fresnel zone with radius y

$\displaystyle\int_0^y e^\frac{-j\beta(z)^2}\rho dz$ , where j is complex and $\beta$ and $\rho$ are constants.

I have seen other questions such as

$\int e^{ix^2}dx$ Complex Integration

but I do not understand how the bounds in that example were obtained when converted to polar coordinates. I have also seen other references that say to use euler's relationship but that leads to a very complicated result using the maclaurin series,

I know I have to use u-substitution to get

$\int_0^y e^{-ju^2}du$

But don't know how to continue. Any help is appreciated!

Answer 1

$$\int_0^y\int_0^{2\pi} e^{-jr^2}rdrd\theta$$

Is only equal to $I^2$ in the limit $y \to \infty$

Notice that the region of integration was a square and has now become a circle.

I'm fairly certain that $ \int_0 ^y e^{-ax^2} dx$ cannot be expressed in terms of elementary functions, its solution is sometimes referred to as the error function.