The question asks to:

1. Prove that $\Gamma(n+1) = n \cdot \Gamma(n)$ for $n > 0$, using the recurrence relation of the Gamma function.
2. Compute $\Gamma(5/2)$ using this recurrence relation.

Let’s solve this step-by-step:

Proof of the Recurrence Relation:

The Gamma function is defined as: $\Gamma(x) = \int_0^\infty t^{x-1} e^{-t} \, dt$

If we substitute $x = n+1$ into the definition, we have: $\Gamma(n+1) = \int_0^\infty t^{n} e^{-t} \, dt$

Using integration by parts:

• Let $u = t^n$, so $du = n t^{n-1} \, dt$
• Let $dv = e^{-t} \, dt$, so $v = -e^{-t}$

Then, $\int t^n e^{-t} \, dt = -t^n e^{-t} \Big|_0^\infty + n \int t^{n-1} e^{-t} \, dt$

The term $-t^n e^{-t} \Big|_0^\infty$ vanishes because:

• As $t \to \infty$, $e^{-t}$ decays faster than $t^n$ grows.
• At $t = 0$, $t^n = 0$.

Thus: $\Gamma(n+1) = n \int_0^\infty t^{n-1} e^{-t} \, dt = n \Gamma(n)$

This proves the recurrence relation: $\Gamma(n+1) = n \cdot \Gamma(n)$

Compute $\Gamma(5/2)$:

Using the recurrence relation $\Gamma(n+1) = n \cdot \Gamma(n)$, we compute step-by-step:

1. Start with $\Gamma(5/2)$: Using the recurrence, $\Gamma(5/2) = \frac{3}{2} \cdot \Gamma(3/2)$.

2. Now compute $\Gamma(3/2)$: Again using the recurrence, $\Gamma(3/2) = \frac{1}{2} \cdot \Gamma(1/2)$.

3. Evaluate $\Gamma(1/2)$: It is a known fact that $\Gamma(1/2) = \sqrt{\pi}$.

4. Substitute back: $\Gamma(3/2) = \frac{1}{2} \cdot \sqrt{\pi}$

   $\Gamma(5/2) = \frac{3}{2} \cdot \frac{1}{2} \cdot \sqrt{\pi} = \frac{3}{4} \sqrt{\pi}$

Thus, $\Gamma(5/2) = \frac{3}{4} \sqrt{\pi}$.

Final Results:

1. The recurrence relation is proved: $\Gamma(n+1) = n \cdot \Gamma(n)$.
2. The value of $\Gamma(5/2)$ is $\frac{3}{4} \sqrt{\pi}$.

Would you like a more detailed explanation or further derivations?
Here are 5 questions to extend your understanding:

1. What is the significance of the Gamma function in mathematics and physics?
2. How is $\Gamma(x)$ related to factorials for integer values of $x$?
3. Can you use the recurrence relation to compute $\Gamma(7/2)$?
4. How would you prove that $\Gamma(1) = 1$ using the definition of $\Gamma(x)$?
5. Can you explore the relationship between the Gamma function and the Beta function?

Tip: When working with the Gamma function, remember its symmetry properties and relation to factorials, as these often simplify calculations.