Show that the Gamma function Gamma(p) = integral from 0

Show that the Gamma function $$ \Gamma(p) = \int_0^\infty x^{p-1} e^{-x} \, dx $$ converges for $p > 0$.

The Gamma function is improper at both $x = 0$ and $x = \infty$. We must analyze convergence at both points.

Step 1: Split the integral

We split at $x = 1$ to separate the potential singularity at 0 from the behavior at infinity:

$$\Gamma(p) = \int_0^1 x^{p-1} e^{-x} \, dx + \int_1^\infty x^{p-1} e^{-x} \, dx$$

For convergence, BOTH integrals must converge.

Step 2: Analyze convergence near $x = 0$

Consider $\int_0^1 x^{p-1} e^{-x} \, dx$.

As $x \to 0^+$:

• $x^{p-1} \to \begin{cases} 0 & \text{if } p > 1 \\ 1 & \text{if } p = 1 \\ +\infty & \text{if } p < 1 \end{cases}$
• $e^{-x} \to e^0 = 1$ (bounded)

Therefore, near $x = 0$:

$$x^{p-1} e^{-x} \approx x^{p-1} \cdot 1 = x^{p-1}$$

Convergence criterion at $x = 0$:

The integral $\int_0^1 x^{p-1} \, dx$ is of the form $\int_0^a x^\alpha \, dx$.

We know this converges if and only if $\alpha > -1$.

In our case, $\alpha = p - 1$, so we need:

$$p - 1 > -1 \implies p > 0$$

Rigorous comparison:

For $x \in (0, 1]$, we have $e^{-x} \leq 1$, so:

$$0 \leq x^{p-1} e^{-x} \leq x^{p-1}$$

Since $\int_0^1 x^{p-1} \, dx$ converges for $p > 0$, by the comparison test:

$$\int_0^1 x^{p-1} e^{-x} \, dx \quad \text{converges for } p > 0$$

Therefore: The integral converges at $x = 0$ if and only if **$p > 0$**. ✓

Step 3: Analyze convergence at $x = \infty$

Consider $\int_1^\infty x^{p-1} e^{-x} \, dx$.

As $x \to \infty$:

• $x^{p-1} \to +\infty$ (polynomial growth)
• $e^{-x} \to 0$ (exponential decay)

The key question: Which dominates?

Fundamental fact: Exponential decay dominates polynomial growth!

For any $n > 0$, we have:

$$\lim_{x \to \infty} x^n e^{-x} = 0$$

More precisely, for large $x$:

$$x^{p-1} e^{-x} < e^{-x/2}$$

for $x$ sufficiently large (independent of $p$).

Proof of exponential dominance:

For any $p$, choose $N$ large enough so that for $x \geq N$:

$$x^{p-1} < e^{x/2}$$

This is true for large enough $x$ since exponential growth beats polynomial growth.

Then for $x \geq N$:

$$x^{p-1} e^{-x} < e^{x/2} e^{-x} = e^{-x/2}$$

Since $\int_N^\infty e^{-x/2} \, dx = 2e^{-N/2}$ converges, by comparison:

$$\int_N^\infty x^{p-1} e^{-x} \, dx \quad \text{converges}$$

And since $\int_1^N x^{p-1} e^{-x} \, dx$ is a proper integral (finite), we conclude:

$$\int_1^\infty x^{p-1} e^{-x} \, dx \quad \text{converges for all } p$$

Therefore: The integral converges at $x = \infty$ **for all $p$** (no restriction needed). ✓

Step 4: Conclusion

Combining both conditions:

• Convergence at $x = 0$: requires $p > 0$
• Convergence at $x = \infty$: holds for all $p$

Convergence of Gamma Function

The Gamma function

$$\Gamma(p) = \int_0^\infty x^{p-1} e^{-x} \, dx$$

converges if and only if $p > 0$.

$$\boxed{\text{For } p > 0, \text{ the Gamma function converges}}$$

Alternative proof for convergence at infinity

We can be more explicit. For $x \geq 1$:

$$x^{p-1} e^{-x} = \frac{x^{p-1}}{e^x}$$

By L'Hôpital's rule applied repeatedly (or by Taylor series), for any $n$:

$$\lim_{x \to \infty} \frac{x^n}{e^x} = 0$$

Therefore, for large $x$:

$$\frac{x^{p-1}}{e^x} < \frac{1}{x^2}$$

Since $\int_1^\infty \frac{1}{x^2} \, dx = 1$ converges, our integral converges at infinity.

Special values and properties

Property 1: $\Gamma(1) = 1$

$$\Gamma(1) = \int_0^\infty e^{-x} \, dx = [-e^{-x}]_0^\infty = 0 - (-1) = 1$$

Property 2: Recurrence relation

For $p > 0$:

$$\Gamma(p+1) = p\Gamma(p)$$

Proof by integration by parts:

Let $u = x^p$ and $dv = e^{-x} \, dx$.

Then $du = px^{p-1} \, dx$ and $v = -e^{-x}$.

$$\Gamma(p+1) = \int_0^\infty x^p e^{-x} \, dx = [-x^p e^{-x}]_0^\infty + \int_0^\infty px^{p-1} e^{-x} \, dx$$

The boundary term vanishes:

• At $x = 0$: $x^p e^{-x} \to 0$ (for $p > 0$)
• At $x = \infty$: $x^p e^{-x} \to 0$ (exponential decay)

Therefore:

$$\Gamma(p+1) = 0 + p\int_0^\infty x^{p-1} e^{-x} \, dx = p\Gamma(p)$$

Property 3: For positive integers

$$\Gamma(n) = (n-1)! \quad \text{for } n \in \mathbb{N}$$

Proof by induction:

Base case: $\Gamma(1) = 1 = 0!$ ✓

Inductive step: If $\Gamma(n) = (n-1)!$, then:

$$\Gamma(n+1) = n\Gamma(n) = n \cdot (n-1)! = n!$$

Property 4: $\Gamma(1/2) = \sqrt{\pi}$

This can be proved using the substitution $x = t^2$ and the Gaussian integral.

Numerical examples

Example 1: $p = 1$

$$\Gamma(1) = 1$$

Example 2: $p = 2$

$$\Gamma(2) = 1 \cdot \Gamma(1) = 1 = 1!$$

Example 3: $p = 3$

$$\Gamma(3) = 2 \cdot \Gamma(2) = 2 \cdot 1 = 2 = 2!$$

Example 4: $p = 1/2$

$$\Gamma(1/2) = \sqrt{\pi} \approx 1.772$$

Example 5: $p = 0.1$

$$\Gamma(0.1) \approx 9.513$$

(Note: $\Gamma(p)$ is very large for small positive $p$)

Why $p > 0$ is necessary

For $p \leq 0$:

Near $x = 0$, the integrand $x^{p-1} e^{-x} \approx x^{p-1}$.

If $p \leq 0$, then $p - 1 \leq -1$, and:

$$\int_0^1 x^{p-1} \, dx \quad \text{diverges}$$

Therefore, $\Gamma(p)$ does not converge for $p \leq 0$.

Applications

The Gamma function appears in:

• Probability theory: Gamma distribution, Chi-squared distribution
• Combinatorics: Generalization of factorials to real/complex numbers
• Physics: Quantum mechanics, statistical mechanics
• Special functions: Bessel functions, hypergeometric functions
• Number theory: Analytic continuation, Riemann zeta function

Summary

1. The Gamma function has potential issues at $x = 0$ and $x = \infty$
2. Near $x = 0$: integrand $\sim x^{p-1}$, converges if $p > 0$
3. At $x = \infty$: exponential decay $e^{-x}$ dominates any polynomial growth
4. Combined: convergence requires only $p > 0$
5. For positive integers: $\Gamma(n) = (n-1)!$
6. Recurrence relation: $\Gamma(p+1) = p\Gamma(p)$

Key lessons:

• Exponential decay ensures convergence at infinity regardless of polynomial factors
• The condition $p > 0$ comes entirely from the behavior at $x = 0$
• The Gamma function extends the factorial to all positive real numbers
• Integration by parts reveals the recurrence relation
• The comparison test with exponentials is essential for proving convergence at infinity
• This is one of the most important special functions in mathematics