Q-Function representation in Mathematica

Question

I observed that there is no Q-function representation in Mathematica. The definition of Q-Function is:

\begin{align} Q(x) &= \frac{1}{\sqrt{2\pi}} \int_x^\infty e^{-\frac{u^2}{2}}du \\ &=\frac{1}{2} \mathrm{erfc}(\frac{x}{\sqrt{2}}) \end{align}

a nother representation of Q-function is: \begin{align} Q(x) = \frac{1}{\pi} \int_0^{\frac{\pi}{2}} e^{\left(\frac{-x^2}{2\sin^2{\phi}} \right)}d\phi \end{align}

there is only Complementary Error Function representation Erfc.

$\mathrm{erfc}(x) = \frac{2}{\sqrt{\pi}} \int_{x}^{\infty} \exp(-t^2) dt$

Is there any possiblity that define a new function in Mathematica? so that it will be a permanent function.

Note: In statistics, the Q-function is the tail probability of the standard normal distribution. In other words, Q(x) is the probability that a normal (Gaussian) random variable will obtain a value larger than x standard deviations above the mean.

Answer 1

There are two parts to your question: how to define the function and how to make it persist between sessions. I suggested

q[x_] := Erfc[x/Sqrt[2]]/2;

though chuy's suggestion looks fine too.To make it persist, you can place this function definition in an initialization cell, and then this will run whenever the notebook is opened. You can find instructions for this here and other information here about running the cell on startup.

Answer 2

If this the function that you use daily, you can put it into you initialization file. Use the command in a notebook:

$UserBasedDirectory

Usually there is a ./Kernel/init.m file. It is evaluated when Mathematica kernel starts. You can put these line in it:

q::usage="q[x] is my Q-function. You cannot modify it, aha!"
q[x_:0] := Erfc[x/Sqrt[2]]/2
Protect[q]

Then you will have several good features: