How to calculate the erfc of a number

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In summary, the conversation discusses the use of the erfc function in solving a problem related to calculating the junction depth of phosphorus diffusion in a p-type wafer. The person is having trouble finding the erfc function on their calculator and is unsure about its properties. The conversation suggests using the Wikipedia page on the Error Function or using the normal CDF function on the calculator to approximate erfc. It also mentions that solving for x in the equation involving erfc can be done through trial-and-error or using more advanced methods like Newton's Method.
  • #1
vysero
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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?
 
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  • #2
vysero said:
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 they 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?
The Wikipedia page on the Error Function may be helpful to you.
 
  • #3
vysero said:
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?

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.
 

FAQ: How to calculate the erfc of a number

What is the formula for calculating the erfc of a number?

The formula for calculating the erfc of a number is erfc(x) = 1 - erf(x), where erf(x) is the error function.

How do I calculate the erfc of a negative number?

To calculate the erfc of a negative number, you can use the formula erfc(x) = 2 - erfc(-x).

Can I use a calculator to calculate the erfc of a number?

Yes, most calculators have a built-in function for calculating the erfc of a number. However, make sure to check the user manual or online documentation to see how to access this function.

What is the range of values for the erfc function?

The erfc function can return values between 0 and 2, where 0 represents a perfect match and 2 represents no match at all.

How is the erfc of a number related to probability?

The erfc of a number is closely related to the probability of a normal distribution. Specifically, the erfc(x) value represents the probability that a normally distributed variable will fall outside the range of -x to x.

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