Why should $$\int_1^{\infty}\exp(ix^2)dx,\int_1^{\infty}\exp(-ix^2)dx,\int_1^{\infty}\exp(-x^2)dx$$ converges but not: $$\int_1^{\infty}\exp(x^2)dx$$
Is there any way that assigns a value to $\int_1^{\infty}\exp(x^2)dx$ which is consistent with the three former? (similar to $1+2+3+...=-\frac{1}{12}$)
EDIT: By suggestions from Nate, I think I should point out that I'm interested in the analytical continuation of $z\rightarrow \int_1^{\infty}\exp(zx^2)dx$. Does it exist? Does it have application in other fields?
Why? Because the integrands of the first two are oscillating trigonometric functions, whose positive parts cancel out their negative ones in such a way that the final result is finite. $~$ The third one is a powerfully decreasing, and therefore convergent, exponential function. The last one is a powerfully increasing exponential function, diverging towards infinity.