Question

The value of \(\Gamma\left(\frac{1}{2}\right)\) is:

• A. \(\pi\)
• B. \(\frac{\pi}{4}\)
• C. \(\sqrt{\pi}\)
• D. \(\frac{\pi}{2}\)

Hint:

Memorize: \(\Gamma(1)=1\), \(\Gamma(n)=(n-1)!\), and \(\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}\). These appear very frequently in exams.

Answer 1

The Correct Option is C

Concept: The Gamma function is an extension of the factorial function to non-integer values and is defined by: \[ \Gamma(n) = \int_0^\infty x^{n-1} e^{-x} dx \] A very important special value is: \[ \Gamma\left(\frac{1}{2}\right) \]

Step 1: Understand its connection with Gaussian integral.
A fundamental result in calculus is: \[ \int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi} \]

Step 2: Transform Gamma function into Gaussian form.
By using substitution \(x = t^2\), the Gamma function becomes: \[ \Gamma\left(\frac{1}{2}\right) = \int_0^\infty x^{-1/2} e^{-x} dx \] After transformation and simplification, it evaluates to: \[ \sqrt{\pi} \]

Step 3: Final conclusion.
\[ \Gamma\left(\frac{1}{2}\right) = \sqrt{\pi} \] Final Answer: \[ \boxed{\sqrt{\pi}} \]