In the article we define partial functions. We also define the following notions related to partial functions and functions themselves: the empty function, the restriction of a function to a partial function from a set into a set, the set of all partial functions from a set into a set, the total functions, the relation of tolerance of two functions and the set of all total functions which are t...

The partial theta function is the sum of series \medskip\centerline{$\displaystyle\theta (q,x):=\sum\nolimits _{j=0}^{\infty}q^{j(j+1)/2}x^j$,}\medskip\noi where $q$is a real or complex parameter ($|q|<1$). Its name due to similaritieswith formula for Jacobi function$\Theta (q,x):=\sum _{j=-\infty}^{\infty}q^{j^2}x^j$. $\theta$ has been considered in Ramanujan's lost notebook. Itfinds applic...