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<center><b><big><big>Arccosine Function Forward Taylor Polynomial Theory</big></big></b></center>
If
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>F</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<mi>arccos</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>
it follows that
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<msqrt><mrow><mn>1</mn>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>x</mi>
</mrow>
</msqrt>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>F</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">-</mo>
<mn>0</mn>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>F</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<mn>-1</mn>
</mrow></math>
and in the
<a href="forwardtheory.xml#Standard Math Functions.Differential Equation" target="_top"><span style='white-space: nowrap'>standard math function differential equation</span></a>
,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>A</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow></math>
,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>B</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<msqrt><mrow><mn>1</mn>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>x</mi>
</mrow>
</msqrt>
</mrow></math>
,
and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>D</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<mn>-1</mn>
</mrow></math>
.
We use
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>a</mi>
</mrow></math>
,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>b</mi>
</mrow></math>
,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>d</mi>
</mrow></math>
and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>z</mi>
</mrow></math>
to denote the
Taylor coefficients for
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>A</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mrow></math>
,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>B</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mrow></math>
,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>D</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mrow></math>
,
and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>F</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mrow></math>
respectively.
<code><span style='white-space: nowrap'><br/>
<br/>
</span></code>We define
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>Q</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<mn>1</mn>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>x</mi>
</mrow></math>
and let
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>q</mi>
</mrow></math>
be the corresponding Taylor coefficients for
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>Q</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mrow></math>
.
It follows that
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<msup><mi mathvariant='italic'>q</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">=</mo>
<mrow><mo stretchy="true">{</mo><mrow><mtable rowalign="center" ><mtr><mtd columnalign="left" >
<mn>1</mn>
<mo stretchy="false">-</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mtd><mtd columnalign="left" >
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>if</mi>
</mstyle></mrow>
<mspace width='.3em'/>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mtd></mtr><mtr><mtd columnalign="left" >
<mo stretchy="false">-</mo>
<munderover><mo displaystyle='true' largeop='true'>∑</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow>
<mi mathvariant='italic'>j</mi>
</munderover>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mtd><mtd columnalign="left" >
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>otherwise</mi>
</mstyle></mrow>
</mtd></mtr></mtable>
</mrow><mo stretchy="true"> </mo></mrow>
</mrow></math>
It follows that
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>B</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
<mo stretchy="false">=</mo>
<msqrt><mrow><mi mathvariant='italic'>Q</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mrow>
</msqrt>
</mrow></math>
and
from the equations for the
<a href="sqrtforward.xml" target="_top"><span style='white-space: nowrap'>square root</span></a>
that for
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mn>1</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">…</mo>
</mrow></math>
,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<msup><mi mathvariant='italic'>b</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msqrt><mrow><msup><mi mathvariant='italic'>q</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</msqrt>
</mtd></mtr><mtr><mtd columnalign="right" >
<msup><mi mathvariant='italic'>b</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mn>1</mn>
</mrow>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mfrac><mrow><mn>1</mn>
</mrow>
<mrow><msup><mi mathvariant='italic'>b</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
<mrow><mo stretchy="true">(</mo><mrow><mfrac><mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
</mrow>
<mrow><mn>2</mn>
</mrow>
</mfrac>
<msup><mi mathvariant='italic'>q</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">-</mo>
<munderover><mo displaystyle='true' largeop='true'>∑</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">=</mo>
<mn>1</mn>
</mrow>
<mi mathvariant='italic'>j</mi>
</munderover>
<mi mathvariant='italic'>k</mi>
<msup><mi mathvariant='italic'>b</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<msup><mi mathvariant='italic'>b</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow><mo stretchy="true">)</mo></mrow>
</mtd></mtr></mtable>
</mrow></math>
It now follows from the general
<a href="forwardtheory.xml#Standard Math Functions.Taylor Coefficients Recursion Formula" target="_top"><span style='white-space: nowrap'>Taylor coefficients recursion formula</span></a>
that for
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mn>1</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">…</mo>
</mrow></math>
,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mi>arccos</mi>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mtd></mtr><mtr><mtd columnalign="right" >
<msup><mi mathvariant='italic'>e</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msup><mi mathvariant='italic'>d</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">+</mo>
<munderover><mo displaystyle='true' largeop='true'>∑</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow>
<mrow><mi mathvariant='italic'>j</mi>
</mrow>
</munderover>
<msup><mi mathvariant='italic'>a</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mtd></mtr><mtr><mtd columnalign="right" >
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mrow><mo stretchy="true">{</mo><mrow><mtable rowalign="center" ><mtr><mtd columnalign="left" >
<mn>-1</mn>
</mtd><mtd columnalign="left" >
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>if</mi>
</mstyle></mrow>
<mspace width='.3em'/>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mtd></mtr><mtr><mtd columnalign="left" >
<mn>0</mn>
</mtd><mtd columnalign="left" >
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>otherwise</mi>
</mstyle></mrow>
</mtd></mtr></mtable>
</mrow><mo stretchy="true"> </mo></mrow>
</mtd></mtr><mtr><mtd columnalign="right" >
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mn>1</mn>
</mrow>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mfrac><mrow><mn>1</mn>
</mrow>
<mrow><msup><mi mathvariant='italic'>b</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
<mrow><mo stretchy="true">(</mo><mrow><munderover><mo displaystyle='true' largeop='true'>∑</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow>
<mi mathvariant='italic'>j</mi>
</munderover>
<msup><mi mathvariant='italic'>e</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">-</mo>
<munderover><mo displaystyle='true' largeop='true'>∑</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">=</mo>
<mn>1</mn>
</mrow>
<mi mathvariant='italic'>j</mi>
</munderover>
<msup><mi mathvariant='italic'>b</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow><mo stretchy="true">)</mo></mrow>
</mtd></mtr><mtr><mtd columnalign="right" >
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mo stretchy="false">-</mo>
<mfrac><mrow><mn>1</mn>
</mrow>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mfrac><mrow><mn>1</mn>
</mrow>
<mrow><msup><mi mathvariant='italic'>b</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
<mrow><mo stretchy="true">(</mo><mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">+</mo>
<munderover><mo displaystyle='true' largeop='true'>∑</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">=</mo>
<mn>1</mn>
</mrow>
<mi mathvariant='italic'>j</mi>
</munderover>
<mi mathvariant='italic'>k</mi>
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<msup><mi mathvariant='italic'>b</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow><mo stretchy="true">)</mo></mrow>
</mtd></mtr></mtable>
</mrow></math>
<hr/>Input File: omh/acos_forward.omh
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</html>
