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Error function erf series

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A common technique for numerical evaluation of functions is to use a strong series for small arguments and an asymptotic series for large arguments. This can be refined using a third approach, such as rational approximation, in the middle.

Error function erf (X) There are alternating series at both ends: the holding series and its asymptotic series are alternating series, so both are examples that fit the previous post.

The error function is also a hypergeometric function, so it is also an example of other things I have written about recently, albeit with a small twist.

The error function is a minor variation of Normal CDF distribution Φ. Analysts prefer to work with erf (X) While statisticians prefer to work with Φ (X). See these notes for translation between the two.

Power series

The power series for the error function is

The series gathers for everyone X, But it’s not practical for adults X. to all X The sentence of the alternating series applies eventually, but for a large X You may have to go far in the series before that happens.

Asymptotic series

The complementary error function erfc (X) Provided by

erfc (X) = 1 – erf (X).

Although the relationship between the two functions is trivial, it is theoretically and numerically useful to get names for the two functions. See this post on mathematical functions that seem unnecessary for further explanation.

The erfc function (X) Has an alternating asymptotic series.

 text erfc (x) =  frac e ^ - x ^ 2  sqrt  pi x  sum_ n = 0 ^  infty  frac (2n-1) !!   2 ^ n x ^ - 2n

Here A!!, A A factorial multiple, is a product of positive integers lower than or equal to A And in the same equality as A. Since in our case A = 2N – 1, A It is odd so we have the product of the positive odd numbers up to and including 2N – 1. The first term in the series includes (-1) !! Defined as 1: There are no positive integers less than 1 or equal to 1, so the double factor of 1 is an empty product, and empty products are defined as 1.

Confluent Hypergeometric Series

The error function is a hypergeometric function, but of a special kind. (As is often the case, this is not really a hypergeometric function but is trivially related to a hypergeometric function.)

The term “hypergeometric function” is overload and means two different things. The most stringent use of the term is for functions and(A, B; third; X) Where are the parameters A and B Specify terms in the coefficient counter and in the parameter third Indicates terms in the denominator. You can find full details here.

The more general use of the term hypergeometric function refers to functions that can have any number of numerator and denominator terms. Again, the full details are here.

The special case of one numerator parameter and one denominator parameter known as a Hypergeometric function merges. Why “Conflict”? The name comes from the use of these functions in solving differential equations. Confluent hyper-geometric functions are suitable for a case where two singularities of a differential equation merge together, like the confluence of two rivers. More about hypergeometric functions and differential equations here.

Without further ado, here are two ways to link the error function to connecting hyper-geometric functions.

 text erf (x) =  frac 2x  sqrt  pi F  left ( frac 1 2;  frac 3 2; -x ^ 2  right ) =  frac 2x  sqrt  pi e ^ - x ^ 2 F  left (1;  frac 3 2; x ^ 2  right)

The middle expression is the same as the series of powers above. The third phrase is new.

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