For questions regarding the theory and evaluation of the Gaussian integral, also known as the Euler–Poisson integral is the integral of the Gaussian function $~e^{−x^2}~$ over the entire real line. . It is named after the German mathematician Carl Friedrich Gauss.
The Gaussian integral or, Euler–Poisson integra or, the probability integral , closely related to the erf function, appears in many situations in engineering mathematics and statistics. It can be defined by
$$I(\alpha)=\int_{-\infty}^{+\infty}e^{-\alpha ~x^2}~ dx$$
Laplace $~(1778)~$ proved that $$\int_{-\infty}^{+\infty}e^{- ~x^2}~ dx=\sqrt{\pi}$$
Applications:
The integral has a wide range of applications. For example, with a slight change of variables it is used to compute the normalizing constant of the normal distribution. The same integral with finite limits is closely related to both the error function and the cumulative distribution function of the normal distribution. In physics this type of integral appears frequently, for example, in quantum mechanics, to find the probability density of the ground state of the harmonic oscillator. This integral is also used in the path integral formulation, to find the propagator of the harmonic oscillator, and in statistical mechanics, to find its partition function.
Some other forms:
$$\int_{-\infty}^{+\infty}e^{-\alpha ~x^2}~ dx=\sqrt{\frac{\pi}{\alpha}}$$
$$\int_{0}^{\infty}e^{-\alpha ~x^2}~ dx=\frac{1}{2}~\sqrt{\frac{\pi}{\alpha}}$$
$$\int_{-\infty}^{+\infty}e^{-\alpha ~x^2+bx}~ dx=e^{\frac{b^2}{4\alpha}}\sqrt{\frac{\pi}{\alpha}}$$
$$\int_{0}^{+\infty}e^{i\alpha ~x^2}~ dx=\frac{1}{2}~\sqrt{\frac{i~\pi}{\alpha}}$$
$$\int_{0}^{+\infty}e^{-i\alpha ~x^2}~ dx=\frac{1}{2}~\sqrt{\frac{\pi}{i~\alpha}}$$
References:
