The Gaussian Integral: Solving and Proving the Most Beautiful Result

The Gaussian integral is a fundamental concept in various fields such as mathematics, physics, and engineering. The integral in question is:

Definition:

Consider the integral:

$$I int_{-infty}^{infty} e^{-x^2} dx$$

Step 1: Square the Integral

One approach to solving this integral is to square it:

$$I^2 left(int_{-infty}^{infty} e^{-x^2} dxright) left(int_{-infty}^{infty} e^{-y^2} dyright)$$

Due to the commutative property of multiplication, we have:

$$I^2 int_{-infty}^{infty} int_{-infty}^{infty} e^{-x^2} e^{-y^2} dx dy$$

Step 2: Simplify the Integral

Notice that:

$$I^2 int_{-infty}^{infty} int_{-infty}^{infty} e^{-(x^2 y^2)} dx dy$$

By converting to polar coordinates, we have:

$$I^2 int_{0}^{2pi} int_{0}^{infty} e^{-r^2} r dr dtheta$$

Step 3: Evaluate the Integral

Notice that the inner integral is:

$$int_{0}^{infty} e^{-r^2} r dr -frac{1}{2} e^{-r^2} bigg|_{0}^{infty} frac{1}{2}$$

Thus, the integral simplifies to:

$$I^2 int_{0}^{2pi} frac{1}{2} dtheta pi$$

Hence:

$$I sqrt{pi}$$

First Method: Using Double Integral

The Gaussian integral can also be solved by a double integral approach in Cartesian coordinates:

$$I^2 int_{-infty}^{infty} int_{-infty}^{infty} e^{-x^2} e^{-y^2} dx dy$$

By converting to polar coordinates:

$$I^2 int_{0}^{2pi} int_{0}^{infty} e^{-r^2} r dr dtheta pi$$

Thus:

$$I sqrt{pi}$$

Second Method: Using Gamma Function

The integral can also be solved using the Gamma function:

Consider:

$$I int_{-infty}^{infty} e^{-lambda^2} dlambda$$

Since $$e^{-lambda^2}$$ is an even function, we can write:

$$I 2 int_{0}^{infty} e^{-lambda^2} dlambda$$

Now, using the substitution $$lambda^2 S Rightarrow 2lambda dlambda dS Rightarrow dlambda frac{1}{2sqrt{S}}dS$$ the integral changes to:

$$I 2int_{0}^{infty} e^{-S} left(frac{1}{2sqrt{S}}right) dS int_{0}^{infty} e^{-S} S^{-frac{1}{2}} dS$$

Recognizing this as the integral representation of the Gamma function:

$$Gamma(n) int_{0}^{infty} e^{-t} t^{n-1} dt$$

We can write:

$$I Gammaleft(frac{1}{2}right) sqrt{pi}$$

Final Result

The Gaussian integral is:

$$int_{-infty}^{infty} e^{-lambda^2} dlambda sqrt{pi}$$

Conclusion

The Gaussian integral is a beautiful and fundamental result, and these methods offer different perspectives on solving it. Understanding these methods not only enhances mathematical proficiency but also provides valuable insights into advanced calculus and probability theory.