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How to calculate the erfc of a number

  1. Sep 30, 2017 #1
    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. jcsd
  3. Sep 30, 2017 #2

    SammyS

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    The Wikipedia page on the Error Function may be helpful to you.
     
  4. Sep 30, 2017 #3

    Ray Vickson

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    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.
     
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