A naive attempt to calculate the Laplace transform of the function $f(t)=e^{t^2}$ results in integrals of the form
$$\int_0^\infty e^{t^2-st}dt,$$ which obviously don't exist as the integrand grows too large. However, Wolfram Alpha/Mathematica gives the following result
$$F\left(\frac{s}{2}\right)-\frac{1}{2} i \sqrt{\pi } e^{-\frac{s^2}{4}} $$ where $F$ is the Dawson function.
Is this really the Laplace transform of $f(t)$? How can it even exist, and why is it complex?
Thank you!