Discussion
The value of \( \Gamma(1/2) \) is
Answer: C
Substituting \( x = t^{2} \) turns the defining integral into a Gaussian:
\[ \Gamma\!\left(\tfrac{1}{2}\right) \;=\; \int_{0}^{\infty} x^{-1/2}e^{-x}dx \;=\; 2\int_{0}^{\infty} e^{-t^{2}}dt \;=\; 2 \cdot \frac{\sqrt{\pi}}{2} \;=\; \sqrt{\pi}. \]
It is the one non-integer argument whose value is met constantly, appearing wherever a Gaussian is integrated
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