# An Elementary Way to Calculate the Gaussian Integral

## Fred Akalin

### January 6, 2011

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:

Exercise 1. (The Gaussian integral.) Let $F(t) = ∫_0^t e^{-x^2} \, dx \text{, }\qquad G(t) = ∫_0^1 \frac{e^{-t^2 (1+x^2)}}{1+x^2} \, dx \text{,}$ 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 $$F(∞) = \displaystyle\lim_{t \to ∞} F(t)$$.
4. 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.

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 $A(t) = ∬\limits_{\triangle_t} e^{-(x^2+y^2)} \, dx \, dy$ 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 $$A(∞) = \lim_{t \to ∞} A(t)$$.
3. Use part b to calculate $∫_{-∞}^{∞} e^{-x^2} \, dx\text{.}$
4. 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 $$G(t)$$.
5. Use part d to derive a proof of part c using only single-variable calculus.

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## Footnotes

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