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erf_forward

Headings-> Derivatives Taylor Coefficients Recursion

@(@\newcommand{\W}[1]{ \; #1 \; } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\B}[1]{ {\bf #1} } \newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} } \newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} } \newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} } \newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }@)@ Error Function Forward Taylor Polynomial Theory
Derivatives
Given @(@ X(t) @)@, we define the function @[@ Z(t) = \R{erf}[ X(t) ] @]@ It follows that @[@ \begin{array}{rcl} \R{erf}^{(1)} ( u ) & = & ( 2 / \sqrt{\pi} ) \exp \left( - u^2 \right) \\ Z^{(1)} (t) & = & \R{erf}^{(1)} [ X(t) ] X^{(1)} (t) = Y(t) X^{(1)} (t) \end{array} @]@ where we define the function @[@ Y(t) = \frac{2}{ \sqrt{\pi} } \exp \left[ - X(t)^2 \right] @]@

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 @[@ \begin{array}{rcl} \sum_{k=1}^j k z^{(k)} t^{k-1} & = & \left[ \sum_{k=0}^j y^{(k)} t^k \right] \left[ \sum_{k=1}^j k x^{(k)} t^{k-1} \right] + o( t^{j-1} ) \end{array} @]@ Setting the coefficients of @(@ t^{j-1} @)@ equal, we have @[@ \begin{array}{rcl} j z^{(j)} = \sum_{k=1}^j k x^{(k)} y^{(j-k)} \\ z^{(j)} = \frac{1}{j} \sum_{k=1}^j k x^{(k)} y^{(j-k)} \end{array} @]@

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Input File: omh/appendix/theory/erf_forward.omh