Cosas varias

 The error function, erf( x ), is defined as the following function: \[ \operatorname{erf}( x) = {2 \over \pi} \int _0 ^x { e^{- t^2} \operatorname{d}t } \tag{1}\] This integral has no analytical solution through any of the conventional functions. Most modern software has a function to evaluate it. However, sometimes it is convenient to have an approximation at hand. It is known that it slowly converges as a series of powers. For this reason, the way to approximate it proposed here is through a geometrically similar function. The proposed base function is: \[y(x) = \tanh \left({{a x}\over 2}\right) = { {2 e^{ a x }} \over { 1 + e^{ a x } }} - 1 \tag{2}\] where \(a\) is a constant to be determined. This particular function is chosen because it satisfies the same function limits; that is to say, \[ \lim_{x \to \infty }{\operatorname{erf}( x)} = \lim_{x \to \infty } { y (x) } = 1 \tag{3.a}\] \[ \lim_{x \to -\infty } {\operatorname{erf}( x)} = \lim_{x \to -\infty }{ y (x) } = -1 \tag{...