Wikipedia error about the complementary error function?

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Discussion Overview

The discussion revolves around the interpretation of the double factorial of -1 and its implications in the context of the complementary error function (erfc). Participants explore the mathematical conventions surrounding factorials of negative integers and the validity of certain formulas.

Discussion Character

  • Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant questions the meaning of taking the double factorial of -1, suggesting it may be a typo in the formula provided.
  • Another participant asserts that the convention used implies that (-1)! = 1.
  • A third participant references the double factorial formula for odd integers, indicating that by setting k = 0, one can arrive at (-1)! = 1, suggesting a convention for handling such cases.
  • One participant proposes a formula for erfc as x approaches infinity and seeks confirmation of its correctness.

Areas of Agreement / Disagreement

Participants express differing views on the treatment of (-1)! and whether the original formula is correctly stated. There is no consensus on whether the formula should be changed or how to interpret the factorial of negative integers.

Contextual Notes

The discussion highlights the reliance on conventions in mathematics, particularly regarding factorials of negative integers, and the potential for ambiguity in formulas derived from these conventions.

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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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They clearly intend that (-1)!=1.
 
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.
 
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?
 

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