In Example 12.15 we introduced Euler's gamma function:

\[\Gamma(t)=\int_{(0, \infty)} x^{t-1} e^{-x} \lambda^{1}(d x)\]

Show that \(\Gamma\left(\frac{1}{2}ight)=\sqrt{\pi}\).

Data from example 12.15

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Example 12.15 (Euler's gamma function) The parameter-dependent integral ) := (0.00)*x^- e- x x (dx), (12.13) T(t):= t>0