Error Function Forward Taylor Polynomial Theory
Keywords: error function forward taylor polynomial theory erf derivatives coefficients recursion

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Error Function Forward Taylor Polynomial Theory
Derivatives
Given X ( t ) , we define the function Z ( t ) = erf [ X ( t ) ] It follows that erf ( 1 ) ( u ) = ( 2 / π ) exp (- u 2 ) Z ( 1 ) ( t ) = erf ( 1 ) [ X ( t ) ] X ( 1 ) ( t ) = Y ( t ) X ( 1 ) ( t ) where we define the function Y ( t ) = 2 π exp [- X ( t ) 2 ]
Taylor Coefficients Recursion
Suppose that we are given the Taylor coefficients up to order j for the function X ( t ) and Y ( t ) . We need a formula that computes the coefficient of order j for Z ( t ) . Using the equation above for Z ( 1 ) ( t ) , we have ∑ k = 1 j k z ( k ) t k -1 = [∑ k = 0 j y ( k ) t k ] [∑ k = 1 j k x ( k ) t k -1 ] + o ( t j -1 ) Setting the coefficients of t j -1 equal, we have j z ( j ) = ∑ k = 1 j k x ( k ) y ( j - k ) z ( j ) = 1 j ∑ k = 1 j k x ( k ) y ( j - k )

Input File: omh/theory/erf_forward.omh