ErrorFunctionWiki[1] - Error function - Wikipedia, the free...

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Error function Plot of the error function From Wikipedia, the free encyclopedia In mathematics, the error function (also called the Gauss error function ) is a special function (non-elementary) which occurs in probability, statistics, materials science, and partial differential equations. It is defined as: The complementary error function , denoted erfc , is defined in terms of the error function: The complex error function , denoted w ( x ), (also known as the Faddeeva function) is also defined in terms of the error function: Contents 1 Properties 1.1 Taylor series 1.2 Inverse function 2 Applications 2.1 Asymptotic expansion 2.2 Approximation with elementary functions 3 Related functions 3.1 Generalised error functions 3.2 Iterated integrals of the complementary error function 4 Implementation 5 Table of values 6 See also 7 References 8 External links Properties Error function - Wikipedia, the free encyclopedia 1 of 8 2/2/2009 11:08 PM
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Fig.2. Integrand exp( z 2 ) in the complex z -plane. Fig.3. erf( z ) in the complex z -plane. The error function is odd: Also, for any complex number z one has where z * is the complex conjugate of z . The integrand ƒ = exp( z 2 ) and ƒ = erf( z ) are shown in the complex z -plane in figures 2 and 3. Level of Im( ƒ ) = 0 is shown with a thick green line. Negative integer values of Im( ƒ ) are shown with thick red lines. Positive integer values of are shown with thick blue lines. Intermediate levels of Im( ƒ ) = constant are shown with thin green lines. Intermediate levels of Re( ƒ ) = constant are shown with thin red lines for negative values and with thin blue lines for positive values. At the real axis, the erf( z ) approach unity at z + and 1 at z −∞ . At the imaginary axis, it tends to . Taylor series The error function is an entire function; it has no singularities (except that at infinity) and its Taylor expansion always converges. The integral cannot be evaluated in closed form in terms of elementary functions, but by expanding the integrand in a Taylor series, one obtains the Taylor series for the error function as follows: which holds for every complex number z . This result arises from the Taylor series expansion of which is and is then integrated term by term. The denominator terms are sequence A007680 in the OEIS.
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This note was uploaded on 11/02/2009 for the course APMA 2102 taught by Professor Keyes during the Spring '08 term at Columbia.

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ErrorFunctionWiki[1] - Error function - Wikipedia, the free...

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