I have learnt several ways to calculate $\int_{0}^{\infty}\cos(t^{2})dt$ and $\int_{0}^{\infty}\sin(t^{2})dt$ before. Now I need to evaluate them by integrating $f(z)=e^{\frac{z^{2}}{2}}$ along the follow contour
I can show that $$ \lim_{R\rightarrow +\infty}\int_{C_{1}(R)}f(z)dz=-i\sqrt{\frac{\pi}{2}} $$ and $$ \lim_{R\rightarrow +\infty}\int_{C_{2}(R)}f(z)dz=(1+i)\int_{0}^{\infty}\cos(t^{2})dt+(-1+i)\int_{0}^{\infty}\sin(t^{2})dt. $$
However, I found it difficult to show $$ \lim_{R\rightarrow +\infty}\int_{C_{3}(R)}f(z)dz=0. $$ Any advice will be helpful. Thanks a lot.
Hint: Write the integral on the segment $C_3$ as $$\left|\int_{C_3}\right|=\left|-\int_0^{R} e^{(x+iR)^2/2}\right|dx\le\int_0^{R}e^{\Re(x^2/2+iRx-R^2/2)}dx=\frac{1}{e^{R^2/2}}\int_0^{R}e^{x^2/2}dx.$$