An Elementary Way to Calculate the Gaussian Integral

Fred Akalin

While reading Timothy Gowers's blog I stumbled on Scott Carnahan's comment describing an elegant calculation of the Gaussian integral \[ \int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}\text{.} \] I was so struck by its elementary character that I imagined what it would be like written up, say, as an extra credit exercise in a single-variable calculus class:

Exercise 1. (The Gaussian integral.) Let \[ F(t) = \int_0^t e^{-x^2} \, dx \text{, }\qquad G(t) = \int_0^1 \frac{e^{-t^2 (1+x^2)}}{1+x^2} \, dx \text{,} \] and \(H(t) = F(t)^2 + G(t)\).

  1. Calculate \(H(0)\).
  2. Calculate and simplify \(H'(t)\). What does this imply about \(H(t)\)?
  3. Use part b to calculate \(F(\infty) = \displaystyle\lim_{t \to \infty} F(t)\).
  4. Use part c to calculate \[ \int_{-\infty}^{\infty} e^{-x^2} \, dx\text{.} \]

Although this is simpler than the usual calculation of the Gaussian integral, for which careful reasoning is needed to justify the use of polar coordinates, it seems more like a certificate than an actual proof; you can convince yourself that the calculation is valid, but you gain no insight into the reasoning that led up to it.[1]

Fortunately, David Speyer's comment solves the mystery; \(G(t)\) falls out of doing the integration in Cartesian coordinates over a triangular region. Just for kicks, here's how I imagine an exercise based on this method would look like (this time for a multi-variable calculus class):

Exercise 2. (The Gaussian integral in Cartesian coordinates.) Let \[ A(t) = \iint\limits_{\triangle_t} e^{-(x^2+y^2)} \, dx \, dy \] where \(\triangle_t\) is the triangle with vertices \((0, 0)\), \((t, 0)\), and \((t, t)\).

  1. Use the substitution \(y = sx\) to reduce \(A(t)\) to a one-dimensional integral.
  2. Use part a to calculate \(A(\infty) = \lim_{t \to \infty} A(t)\).
  3. Use part b to calculate \[ \int_{-\infty}^{\infty} e^{-x^2} \, dx\text{.} \]
  4. Let \[ F(t) = \int_0^t e^{-x^2} \, dx \qquad\text{ and }\qquad G(t) = \int_0^1 \frac{e^{-t^2 (1+x^2)}}{1+x^2} \, dx \text{.} \] Use part a to relate \(F(t)\) to \(G(t)\).
  5. Use part d to derive a proof of part c using only single-variable calculus.

[1] Similar to proving \(\sum\limits_{i=0}^n m^3 = \frac{n^2(n+1)^2}{4}\) by induction.