# How to calculate the erfc of a number

1. Sep 30, 2017

### vysero

So the HW is actually about calculating the junction depth of phosphorus diffusion into a p-type wafer however that is not the problem I am having. The problem I am having is that the book tells me to calculate the erfc(#). However, my calculator does not seem to have an erfc function. In the book the do algebra with the function, for instance it says:

(1.1x10^30)erfc(x/(2sqrt(DT)) = (3x10^16) Solving for x yeilds: x = 2sqrt(DT)erfc^-1(.000273)

What am I missing here? How are they pulling x out of the erfc function? I have never even heard of an erfc function I have no idea what the properties of the function are. For instance if it were exp(x) then I can find x by taking the ln(exp(x)) but how do I do that with erfc?

2. Sep 30, 2017

### SammyS

Staff Emeritus

3. Sep 30, 2017

### Ray Vickson

You can get $\text{erfc}(x)$ in terms of the so-called erf-function: $\text{erfc}(x) = 1 - \text{erf}(x)$. Most calculators lack an "erf" button, but many of them have a "normal distribution" button, giving the cumulative distribution (CDF) of the standard normal distribution. If $\Phi(x)$ is the normal CDF we have
$$\Phi(x) = \frac{1}{2} + \frac{1}{2} \text{erf} \left( \frac{x}{\sqrt{2}} \right)$$
Thus,
$$1-\Phi(\sqrt{2} y) = \frac{1}{2} - \frac{1}{2} \text{erf}(y)= \frac{1}{2} \text{erfc}(y)$$
There are no exact, closed-form formulas for erfc or $\Phi$, but many fast and accurate algorithms are available to compute numerical values reliably, so getting $\Phi(x)$ by pressing a button is really no different from getting $\sin(x)$ by pressing a button.

Note, however, if you want $\text{erfc}^{-1}(z)$ you need to solve the equation $\text{erfc}(x) = z$. You can do that fairly quickly by trial-and-error methods, or by plotting, etc. You could also use fancy techniques like Newton's Method.