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Monthly Archives: March 2019
Integral via Inversions
Let us compute the following integral (1) Performing , this integral becomes (2) Thus we have (3)
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Fresnel integral
The Fresnel integrals read (1) Consider the following integral (2) From this we have (3) which gives (4) Equation (2) also gives (5) We give an alternative derivation in the following. Let us start with … Continue reading
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Integral by Euler’s Formula
We compute the following integral (1) Performing a partial fraction expansion of the integrand (2) Thus we have (3) Let us see another integral (4) Using Euler’s Formula (5) we have (6)
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Gaussian integral
The Gaussian integral is: \begin{align} \int_{\infty}^{\infty}dx e^{x^2} \,=\, \sqrt{\pi} \end{align} A standard way to derive this formula is \begin{align} \bigg(\int_{\infty}^{\infty}dx e^{x^2}\bigg)^2 \,&=\, \int_{\infty}^{\infty}dx \int_{\infty}^{\infty}dy\, e^{(x^2+y^2)} \\ \,&=\, \int_0^{2\pi} d\phi \int_{0}^{\infty}dr\,r e^{r^2} \\ \,&=\, \int_0^{2\pi} d\phi \int_{0}^{\infty}{dr^2 \over 2}\, e^{r^2} \\ … Continue reading
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