Power Series Expansion of the Error Function | Massachusetts Institute of Technology - Edubirdie

University:
Massachusetts Institute of Technology
Course:
18.01SC | Single Variable Calculus
Academic year:

2023
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32

Pages:

1
Author:

lepljen7ewuy

Power Series Expansion of the Error Function Several times in this course we’ve seen the error function: � x 2 2 erf(x) = √ e−t dt. π 0 (The factor of √2π guarantees that limx→∞ erf(x) = 1.) This function is very important in probability theory, but we don’t have a conventional algebraic description of it. Because we can integrate power series and know that: 2 e−t = 1 − t2 + t4 t6 − + ··· 2! 3! (R = ∞), we can now ﬁnd a power series expansion for the error function. � x 2 2 erf(x) = √ e−t dt π 0 � x 2 t4 t6 = √ (1 − t2 + − + · · ·)dt 2! 3! π 0 � �x 3 5 2 t t t7 = √ t− + − + ··· 3 5 · 2! 7 · 3! π 0 � � 2 x3 x5 x7 = √ x− + − + ··· 3 5 · 2! 7 · 3! π To get this to look exactly like a power series we would distribute the factor of √2 across the sum, multiplying it by each term of the series. However, that’s π not strictly necessary. This turns out to be a very good way to compute the value of the error function; your calculator probably uses this method.

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