# 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 \[ ∫_{-∞}^{∞} e^{-x^2} \, dx = \sqrt{π}\text{.} \] 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:

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

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):

- Use the substitution \(y = sx\) to reduce \(A(t)\) to a one-dimensional integral.
- Use part a to calculate \(A(∞) = \lim_{t \to ∞} A(t)\).
- Use part b to calculate \[ ∫_{-∞}^{∞} e^{-x^2} \, dx\text{.} \]
- Let \[ F(t) = ∫_0^t e^{-x^2} \, dx \qquad\text{ and }\qquad G(t) = ∫_0^1 \frac{e^{-t^2 (1+x^2)}}{1+x^2} \, dx \text{.} \] Use part a to relate \(F(t)\) to \(G(t)\).
- Use part d to derive a proof of part c using only single-variable calculus.

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## Footnotes

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