Traditionally we define a gaussian function at a point x (assuming mean to be 0) as follows
$$g_{\sigma}(x) = \frac{1}{\sqrt{2\pi \sigma^{2}}} \exp\left(\frac{x^{2}}{2\sigma^{2}}\right)$$
In some sources however, the exact form is given as follows :
$$g_{\sigma}(x) = \frac{1}{2\pi \sigma^{2}}\exp\left(\frac{x^{2}}{2\sigma^{2}}\right)$$
Why these two forms are used and where which one should be employed ?
For example, when I am working with images, a little difference produces great numerical variations which have an effect in terms of learning.
A gaussian function is any function of the form $$ a\,e^{-b(x-c)^2}. $$ A gaussian probability distribution with mean $0$ and variance $\sigma^2$ corresponds to the first of your definitions.