I have an exercise which seem to be a method of calculating the Gaussian integral :

Let $\displaystyle f(x)=\int_0^1 \frac{\exp(-x^2(t^2+1))}{t^2+1}\ \mathrm{d}t$

Study the deriviability of $f$, then conclude that $\displaystyle \int_0^{\infty} e^{-x^2}\ \mathrm{d}x=\frac{\sqrt{\pi}}{2}$.

I'm stuck in the second question : by Leibinz rule $f$ is differentiable over $\mathbb{R}$, but I couldn't calculate $f'(x)$ in terms of $x$ only or figure out the relation between the two questions.


A few hints to get you going.

Consider $F(t)=\left(\displaystyle \int_0^t e^{-x^2}\ \mathrm{d}x\right)^2$.

Differentiate with respect to $t$ and set $x=ty$. You will arrive at:$$F{\prime}(t)=-{\frac{d}{dt}}\int_0^1 {\frac{e^{-(1+y^2)t^2}}{1+y^2}}\ \mathrm{d}y$$

Write this as: $F^{\prime}(t)=-G^{\prime}(t)$ so there is a constant $C$ such that $F(t)=-G(t)+C$ for all $t>0$.

To find $C$ let $t$ to tent to $0$. The left hand side goes to 0 obviously while the right hand side goes to ${\pi}/4+C$. Hence $C=-{\pi}/4$. Substitute in $F(t)=-G(t)+C$ and let t to tend to infinity to arrive at what you need.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.