Evaluate (Hint: Change variables to convert the integrand to a Gaussian.) Then use the recursion formula to evaluate

To evaluate

Step 1:

For the positive values of , the Gamma function = ----(1)

Evaluating for =

Step 2:

Substituting = in (1)

= ---(2)

= 2

We know that = 1, therefore the above equation become

= 2-----(3)

Step 3:

From the Gaussian integral, using the result of Gaussian integral

=

Splitting the limits of the integral, we get

= + =

Each part is equal and the value for one part is

= -----(4)

Step 4:

Applying the result of (4) in (3)

We get

= 2()

= = 1.7725---(5)

Using Recursion formula

To evaluate

Step 1:

For any positive value of , the gamma function becomes

= (-1) ----(6)

Substituting =

= (-1)

= -----(7)

Step 2:

We know that =

=

= = 0.8862 ----(8)

To evaluate

Step 1:

The previous formula is used only for positive values of . When < 0 the gamma function becomes

= -----(9)

=