Wikipedia error about the complementary error function?

In summary, the conversation discusses taking the double factorial of -1 and whether or not it is a typo. It is noted that the double factorial can be expressed as (2k-1)! = \frac{(2k)!}{2^k k!} for odd integers, and setting k = 0 gives (-1)! = 1. It is also mentioned that the formula can be extended to work for k = 0 by convention. The conversation ends with a question about confirming a formula involving the error function as x approaches infinity.
  • #1
AxiomOfChoice
533
1
Take a trip over here and explain to me what is meant by taking the double factorial of [itex]-1[/itex]. If you try to let [itex]N = 1[/itex] in the remainder formula, you wind up having to take [itex](2(0) - 1)! = (-1)![/itex], right? This strikes me as a typo; should it be changed? If so, to what?
 
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  • #2
They clearly intend that (-1)!=1.
 
  • #3
If you look at the page for double factorial, they note that for odd integers it can be expressed as

[tex](2k-1)! = \frac{(2k)!}{2^k k!}.[/tex]

Setting k = 0 gives (-1)! = 1. The original formula is derived for k > 0, but one can take the formula to work for k = 0 by convention, I guess.
 
  • #4
So I'm guessing that if I set [itex]N = 1[/itex], then what we have is

[tex]
\text{erfc}(x) = \frac{e^{-x^2}}{x \sqrt \pi} + O \left(x e^{-x^2} \right) \quad \text{as $x\to \infty$}.
[/tex]

Can anyone confirm this?
 

1. What is the complementary error function?

The complementary error function, also known as erfc, is a mathematical function that calculates the probability of a normally distributed random variable falling within a certain range.

2. What is the error in Wikipedia's article on the complementary error function?

The error in Wikipedia's article on the complementary error function is that it incorrectly states the formula for calculating erfc. The correct formula is 1 - erf(x).

3. How does the complementary error function relate to the error function?

The complementary error function is the complement of the error function. This means that erfc(x) = 1 - erf(x). It is often used in probability and statistics calculations.

4. Is the Wikipedia error about the complementary error function a common mistake?

Yes, this is a common mistake in mathematical articles and textbooks. However, it is important to use the correct formula, as it can greatly impact the accuracy of calculations.

5. How can I avoid using the incorrect formula for the complementary error function?

To avoid using the incorrect formula, it is important to double check your sources and use reliable and accurate references. Additionally, it is always helpful to consult with a mathematics expert or conduct your own calculations to verify the formula.

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