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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}} }@)@ Inverse Sine and Hyperbolic Sine Forward Mode Theory
Derivatives
@[@ \begin{array}{rcl} \R{asin}^{(1)} (x) & = & 1 / \sqrt{ 1 - x * x } \\ \R{asinh}^{(1)} (x) & = & 1 / \sqrt{ 1 + x * x } \end{array} @]@If @(@ F(x) @)@ is @(@ \R{asin} (x) @)@ or @(@ \R{asinh} (x) @)@ the corresponding derivative satisfies the equation @[@ \sqrt{ 1 \mp x * x } * F^{(1)} (x) - 0 * F (u) = 1 @]@ and in the standard math function differential equation , @(@ A(x) = 0 @)@, @(@ B(x) = \sqrt{1 \mp x * x } @)@, and @(@ D(x) = 1 @)@. We use @(@ a @)@, @(@ b @)@, @(@ d @)@ and @(@ z @)@ to denote the Taylor coefficients for @(@ A [ X (t) ] @)@, @(@ B [ X (t) ] @)@, @(@ D [ X (t) ] @)@, and @(@ F [ X(t) ] @)@ respectively.

Taylor Coefficients Recursion
We define @(@ Q(x) = 1 \mp x * x @)@ and let @(@ q @)@ be the corresponding Taylor coefficients for @(@ Q[ X(t) ] @)@. It follows that @[@ q^{(j)} = \left\{ \begin{array}{ll} 1 \mp x^{(0)} * x^{(0)} & {\rm if} \; j = 0 \\ \mp \sum_{k=0}^j x^{(k)} x^{(j-k)} & {\rm otherwise} \end{array} \right. @]@ It follows that @(@ B[ X(t) ] = \sqrt{ Q[ X(t) ] } @)@ and from the equations for the square root that for @(@ j = 0 , 1, \ldots @)@, @[@ \begin{array}{rcl} b^{(0)} & = & \sqrt{ q^{(0)} } \\ b^{(j+1)} & = & \frac{1}{j+1} \frac{1}{ b^{(0)} } \left( \frac{j+1}{2} q^{(j+1) } - \sum_{k=1}^j k b^{(k)} b^{(j+1-k)} \right) \end{array} @]@ It now follows from the general Taylor coefficients recursion formula that for @(@ j = 0 , 1, \ldots @)@, @[@ \begin{array}{rcl} z^{(0)} & = & F ( x^{(0)} ) \\ e^{(j)} & = & d^{(j)} + \sum_{k=0}^{j} a^{(j-k)} * z^{(k)} \\ & = & \left\{ \begin{array}{ll} 1 & {\rm if} \; j = 0 \\ 0 & {\rm otherwise} \end{array} \right. \\ z^{(j+1)} & = & \frac{1}{j+1} \frac{1}{ b^{(0)} } \left( \sum_{k=0}^j e^{(k)} (j+1-k) x^{(j+1-k)} - \sum_{k=1}^j b^{(k)} (j+1-k) z^{(j+1-k)} \right) \\ z^{(j+1)} & = & \frac{1}{j+1} \frac{1}{ b^{(0)} } \left( (j+1) x^{(j+1)} - \sum_{k=1}^j k z^{(k)} b^{(j+1-k)} \right) \end{array} @]@

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