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[glass snow]

which part are you stuck on?

[sacred dagger]

Let’s solve each part step by step.

Part (a): Compute the integrals using the Gamma function

The Gamma function is defined as:

Comparing with the given integrals:
1. First integral

We rewrite this in the Gamma function form by setting , so . Thus,

Using the property:

Thus,

2.    Second integral

Setting , so , we get:

There is no simple closed-form for , but it can be numerically evaluated or left in terms of the Gamma function.

Part (b): Prove the reflection formula

The reflection formula for the Gamma function states:

Proof:

Using Euler’s reflection formula:
1. Consider the integral representation of the Beta function:

Using the identity:

and setting , , we obtain:

Since , it follows that:

2.    From the known result:

we conclude that:

which proves the reflection formula.

Part (c): Extending Gamma function to complex domain and finding

The Gamma function is extended to complex numbers with using the integral definition:

For :

Using the known result:

Thus:

Part (d): Derive Stirling’s approximation for large

Stirling’s approximation states:

Derivation:
1. Using the integral representation:

For large , the dominant contribution comes from the maximum of , which occurs at .
2. Approximate the integral using the Laplace method:

which simplifies to:

This completes the derivation.