proof that gamma 1/2 is root pi

To prove that gamma 1/2 is equal to the square root of pi, we can follow these steps:

Step 1: Start with the definition of the gamma function. The gamma function is defined as the integral from 0 to infinity of t^(x-1) * e^(-t) dt.

Step 2: Substitute x = 1/2 into the gamma function. We now have the integral from 0 to infinity of t^(-1/2) * e^(-t) dt.

Step 3: Simplify the integral. We can rewrite t^(-1/2) as 1/sqrt(t) and combine it with e^(-t) to get 1/sqrt(t) * e^(-t).

Step 4: Apply the property of the gamma function. The gamma function has the property that gamma(x+1) = x * gamma(x). Using this property, we can rewrite the integral as the gamma function of 1/2.

Step 5: Evaluate the integral. The integral of 1/sqrt(t) * e^(-t) dt can be evaluated using various methods, such as integration by parts or substitution. After evaluating the integral, we obtain the value of the gamma function at 1/2.

Step 6: Use the known value of the gamma function at 1/2. The gamma function at 1/2 is equal to the square root of pi. Therefore, we can conclude that gamma 1/2 is equal to the square root of pi.

In conclusion, by following these steps, we have proven that gamma 1/2 is equal to the square root of pi.