<?xml version='1.0'?>
<html xmlns='http://www.w3.org/1999/xhtml'
      xmlns:math='http://www.w3.org/1998/Math/MathML'
>
<head>
<title>The Theory of Forward Mode</title>
<meta name="description" id="description" content="The Theory of Forward Mode"/>
<meta name="keywords" id="keywords" content=" "/>
<style type='text/css'>
body { color : black }
body { background-color : white }
A:link { color : blue }
A:visited { color : purple }
A:active { color : purple }
</style>
<script type='text/javascript' language='JavaScript' src='_forwardtheory_xml.js'>
</script>
</head>
<body>
<table><tr>
<td>
<a href="http://www.coin-or.org/CppAD/" target="_top"><img border="0" src="_image.gif"/></a>
</td>
<td><a href="theory.xml" target="_top">Prev</a>
</td><td><a href="expforward.xml" target="_top">Next</a>
</td><td>
<select onchange='choose_across0(this)'>
<option>Index-&gt;</option>
<option>contents</option>
<option>reference</option>
<option>index</option>
<option>search</option>
<option>external</option>
</select>
</td>
<td>
<select onchange='choose_up0(this)'>
<option>Up-&gt;</option>
<option>CppAD</option>
<option>Appendix</option>
<option>Theory</option>
<option>ForwardTheory</option>
</select>
</td>
<td>
<select onchange='choose_down3(this)'>
<option>CppAD-&gt;</option>
<option>Install</option>
<option>Introduction</option>
<option>AD</option>
<option>ADFun</option>
<option>preprocessor</option>
<option>multi_thread</option>
<option>library</option>
<option>ipopt_solve</option>
<option>Example</option>
<option>speed</option>
<option>Appendix</option>
</select>
</td>
<td>
<select onchange='choose_down2(this)'>
<option>Appendix-&gt;</option>
<option>Faq</option>
<option>Theory</option>
<option>glossary</option>
<option>Bib</option>
<option>Bugs</option>
<option>WishList</option>
<option>whats_new</option>
<option>deprecated</option>
<option>compare_c</option>
<option>License</option>
</select>
</td>
<td>
<select onchange='choose_down1(this)'>
<option>Theory-&gt;</option>
<option>ForwardTheory</option>
<option>ReverseTheory</option>
<option>reverse_identity</option>
</select>
</td>
<td>
<select onchange='choose_down0(this)'>
<option>ForwardTheory-&gt;</option>
<option>ExpForward</option>
<option>LogForward</option>
<option>SqrtForward</option>
<option>SinCosForward</option>
<option>AtanForward</option>
<option>AsinForward</option>
<option>AcosForward</option>
<option>tan_forward</option>
</select>
</td>
<td>
<select onchange='choose_current0(this)'>
<option>Headings-&gt;</option>
<option>Taylor Notation</option>
<option>Binary Operators</option>
<option>---..Addition</option>
<option>---..Subtraction</option>
<option>---..Multiplication</option>
<option>---..Division</option>
<option>Standard Math Functions</option>
<option>---..Differential Equation</option>
<option>---..Taylor Coefficients Recursion Formula</option>
<option>---..Cases that Apply Recursion Above</option>
<option>---..Special Cases</option>
</select>
</td>
</tr></table><br/>
<center><b><big><big>The Theory of Forward Mode</big></big></b></center>
<br/>
<b><big><a name="Taylor Notation" id="Taylor Notation">Taylor Notation</a></big></b>
<br/>
In Taylor notation, each variable corresponds to 
a function of a single argument which we denote by 
<code><i><font color="black"><span style='white-space: nowrap'>t</span></font></i></code>

(see Section 10.2 of 
<a href="bib.xml#Evaluating Derivatives" target="_top"><span style='white-space: nowrap'>Evaluating&#xA0;Derivatives</span></a>
).
Here and below

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mrow></math>

, 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>Y</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mrow></math>

, and 
<code><i><font color="black"><span style='white-space: nowrap'>Z(t)</span></font></i></code>
 are scalar valued functions 
and the corresponding <i>p</i>-th order Taylor coefficients row vectors are

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>x</mi>
</mrow></math>

, 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>y</mi>
</mrow></math>

 and 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>z</mi>
</mrow></math>

; i.e.,

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="left" >
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="right" >
<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>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">+</mo>
<mo stretchy="false">&#x022EF;</mo>
<mo stretchy="false">+</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>p</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>p</mi>
</msup>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>o</mi>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>p</mi>
</msup>
<mo stretchy="false">)</mo>
</mtd></mtr><mtr><mtd columnalign="left" >
<mi mathvariant='italic'>Y</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="right" >
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">+</mo>
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">+</mo>
<mo stretchy="false">&#x022EF;</mo>
<mo stretchy="false">+</mo>
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>p</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>p</mi>
</msup>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>o</mi>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>p</mi>
</msup>
<mo stretchy="false">)</mo>
</mtd></mtr><mtr><mtd columnalign="left" >
<mi mathvariant='italic'>Z</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="right" >
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">+</mo>
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">+</mo>
<mo stretchy="false">&#x022EF;</mo>
<mo stretchy="false">+</mo>
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>p</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>p</mi>
</msup>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>o</mi>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>p</mi>
</msup>
<mo stretchy="false">)</mo>
</mtd></mtr></mtable>
</mrow></math>

For the purposes of this section, we are given 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>x</mi>
</mrow></math>

 and 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>y</mi>
</mrow></math>


and need to determine 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>z</mi>
</mrow></math>

.

<br/>
<br/>
<b><big><a name="Binary Operators" id="Binary Operators">Binary Operators</a></big></b>


<br/>
<br/>
<b><a name="Binary Operators.Addition" id="Binary Operators.Addition">Addition</a></b>


<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mi mathvariant='italic'>Z</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>Y</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mtd></mtr><mtr><mtd columnalign="right" >
<munderover><mo displaystyle='true' largeop='true'>&#x02211;</mo>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow>
<mi mathvariant='italic'>p</mi>
</munderover>
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>j</mi>
</msup>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<munderover><mo displaystyle='true' largeop='true'>&#x02211;</mo>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow>
<mi mathvariant='italic'>p</mi>
</munderover>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>j</mi>
</msup>
<mo stretchy="false">+</mo>
<munderover><mo displaystyle='true' largeop='true'>&#x02211;</mo>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow>
<mi mathvariant='italic'>p</mi>
</munderover>
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>j</mi>
</msup>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>o</mi>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>p</mi>
</msup>
<mo stretchy="false">)</mo>
</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>
</mrow>
</msup>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">+</mo>
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mtd></mtr></mtable>
</mrow></math>

<br/>
<b><a name="Binary Operators.Subtraction" id="Binary Operators.Subtraction">Subtraction</a></b>


<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mi mathvariant='italic'>Z</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>Y</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mtd></mtr><mtr><mtd columnalign="right" >
<munderover><mo displaystyle='true' largeop='true'>&#x02211;</mo>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow>
<mi mathvariant='italic'>p</mi>
</munderover>
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>j</mi>
</msup>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<munderover><mo displaystyle='true' largeop='true'>&#x02211;</mo>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow>
<mi mathvariant='italic'>p</mi>
</munderover>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>j</mi>
</msup>
<mo stretchy="false">-</mo>
<munderover><mo displaystyle='true' largeop='true'>&#x02211;</mo>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow>
<mi mathvariant='italic'>p</mi>
</munderover>
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>j</mi>
</msup>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>o</mi>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>p</mi>
</msup>
<mo stretchy="false">)</mo>
</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>
</mrow>
</msup>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">-</mo>
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mtd></mtr></mtable>
</mrow></math>

<br/>
<b><a name="Binary Operators.Multiplication" id="Binary Operators.Multiplication">Multiplication</a></b>


<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mi mathvariant='italic'>Z</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>Y</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mtd></mtr><mtr><mtd columnalign="right" >
<munderover><mo displaystyle='true' largeop='true'>&#x02211;</mo>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow>
<mi mathvariant='italic'>p</mi>
</munderover>
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>j</mi>
</msup>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mrow><mo stretchy="true">(</mo><mrow><munderover><mo displaystyle='true' largeop='true'>&#x02211;</mo>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow>
<mi mathvariant='italic'>p</mi>
</munderover>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>j</mi>
</msup>
</mrow><mo stretchy="true">)</mo></mrow>
<mo stretchy="false">*</mo>
<mrow><mo stretchy="true">(</mo><mrow><munderover><mo displaystyle='true' largeop='true'>&#x02211;</mo>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow>
<mi mathvariant='italic'>p</mi>
</munderover>
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>j</mi>
</msup>
</mrow><mo stretchy="true">)</mo></mrow>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>o</mi>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>p</mi>
</msup>
<mo stretchy="false">)</mo>
</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>
</mrow>
</msup>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<munderover><mo displaystyle='true' largeop='true'>&#x02211;</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'>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'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mtd></mtr></mtable>
</mrow></math>

<br/>
<b><a name="Binary Operators.Division" id="Binary Operators.Division">Division</a></b>


<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mi mathvariant='italic'>Z</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">/</mo>
<mi mathvariant='italic'>Y</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mtd></mtr><mtr><mtd columnalign="right" >
<mi mathvariant='italic'>x</mi>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mi mathvariant='italic'>z</mi>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>y</mi>
</mtd></mtr><mtr><mtd columnalign="right" >
<munderover><mo displaystyle='true' largeop='true'>&#x02211;</mo>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow>
<mi mathvariant='italic'>p</mi>
</munderover>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>j</mi>
</msup>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mrow><mo stretchy="true">(</mo><mrow><munderover><mo displaystyle='true' largeop='true'>&#x02211;</mo>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow>
<mi mathvariant='italic'>p</mi>
</munderover>
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>j</mi>
</msup>
</mrow><mo stretchy="true">)</mo></mrow>
<mo stretchy="false">*</mo>
<mrow><mo stretchy="true">(</mo><mrow><munderover><mo displaystyle='true' largeop='true'>&#x02211;</mo>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow>
<mi mathvariant='italic'>p</mi>
</munderover>
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>j</mi>
</msup>
</mrow><mo stretchy="true">)</mo></mrow>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>o</mi>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>p</mi>
</msup>
<mo stretchy="false">)</mo>
</mtd></mtr><mtr><mtd columnalign="right" >
<msup><mi mathvariant='italic'>x</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" >
<munderover><mo displaystyle='true' largeop='true'>&#x02211;</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'>z</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>
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</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>
</mrow>
</msup>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mn>1</mn>
</mrow>
<mrow><msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
<mrow><mo stretchy="true">(</mo><mrow><msup><mi mathvariant='italic'>x</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'>&#x02211;</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'>z</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>
<msup><mi mathvariant='italic'>y</mi>
<mrow><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>

<br/>
<b><big><a name="Standard Math Functions" id="Standard Math Functions">Standard Math Functions</a></big></b>
<br/>
Suppose that 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>F</mi>
</mrow></math>

 is a standard math function and

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>Z</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<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>

<br/>
<b><a name="Standard Math Functions.Differential Equation" id="Standard Math Functions.Differential Equation">Differential Equation</a></b>
<br/>
All of the standard math functions
satisfy a differential equation of the form

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>B</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">)</mo>
<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'>u</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>A</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">)</mo>
<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>
<mi mathvariant='italic'>D</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">)</mo>
</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>

 and 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>d</mi>
</mrow></math>

 to denote the 
<i>p</i>-th order Taylor coefficient row vectors 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>

 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>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mrow></math>


respectively.
We assume that these coefficients are known functions of 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>x</mi>
</mrow></math>

, 
the <i>p</i>-th order Taylor coefficients for 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mrow></math>

.

<br/>
<br/>
<b><a name="Standard Math Functions.Taylor Coefficients Recursion Formula" id="Standard Math Functions.Taylor Coefficients Recursion Formula">Taylor Coefficients Recursion Formula</a></b>
<br/>
Our problem here is to express 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>z</mi>
</mrow></math>

, 
the <i>p</i>-th order Taylor coefficient row vector for 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>Z</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mrow></math>

,
in terms of these other known coefficients.
It follows from the formulas above that

<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>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<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>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>X</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mtd></mtr><mtr><mtd columnalign="right" >
<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>
<msup><mi mathvariant='italic'>Z</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mo stretchy="false">{</mo>
<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>
<mo stretchy="false">+</mo>
<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>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>Z</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">}</mo>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>X</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mtd></mtr><mtr><mtd columnalign="right" >
<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>
<msup><mi mathvariant='italic'>Z</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mi mathvariant='italic'>E</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>X</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mtd></mtr></mtable>
</mrow></math>

where we define

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>E</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<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>
<mo stretchy="false">+</mo>
<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>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>Z</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mrow></math>

We can compute the value of 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow></math>

 using the formula

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>F</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>
</mrow></math>

Suppose by induction (on 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>j</mi>
</mrow></math>

) that we are given the 
Taylor coefficients of 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>E</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">)</mo>
</mrow></math>

 up to order 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>j</mi>
<mn>-1</mn>
</mrow></math>

; i.e.,

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>e</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow></math>

 for 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">&#x02026;</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>j</mi>
<mn>-1</mn>
</mrow></math>


and the coefficients 

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow></math>

 for 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">&#x02026;</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>j</mi>
</mrow></math>

.
We can compute 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>e</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow></math>

 using the formula 

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<msup><mi mathvariant='italic'>e</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">=</mo>
<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'>&#x02211;</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'>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>
</mrow></math>

We need to complete the induction by finding formulas for 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<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>
</mrow></math>

. 
It follows for the formula for the 
<a href="forwardtheory.xml#Binary Operators.Multiplication" target="_top"><span style='white-space: nowrap'>multiplication</span></a>
 
operator that

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mrow><mo stretchy="true">(</mo><mrow><munderover><mo displaystyle='true' largeop='true'>&#x02211;</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'>b</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>k</mi>
</msup>
</mrow><mo stretchy="true">)</mo></mrow>
<mo stretchy="false">*</mo>
<mrow><mo stretchy="true">(</mo><mrow><munderover><mo displaystyle='true' largeop='true'>&#x02211;</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">=</mo>
<mn>1</mn>
</mrow>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
</mrow>
</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>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>t</mi>
<mrow><mi mathvariant='italic'>k</mi>
<mn>-1</mn>
</mrow>
</msup>
</mrow><mo stretchy="true">)</mo></mrow>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mrow><mo stretchy="true">(</mo><mrow><munderover><mo displaystyle='true' largeop='true'>&#x02211;</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>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>k</mi>
</msup>
</mrow><mo stretchy="true">)</mo></mrow>
<mo stretchy="false">*</mo>
<mrow><mo stretchy="true">(</mo><mrow><munderover><mo displaystyle='true' largeop='true'>&#x02211;</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">=</mo>
<mn>1</mn>
</mrow>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
</mrow>
</munderover>
<mi mathvariant='italic'>k</mi>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">*</mo>
<msup><mi mathvariant='italic'>t</mi>
<mrow><mi mathvariant='italic'>k</mi>
<mn>-1</mn>
</mrow>
</msup>
</mrow><mo stretchy="true">)</mo></mrow>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>o</mi>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>p</mi>
</msup>
<mo stretchy="false">)</mo>
</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'>&#x02211;</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'>&#x02211;</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" >
<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'>&#x02211;</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">=</mo>
<mn>1</mn>
</mrow>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
</mrow>
</munderover>
<mi mathvariant='italic'>k</mi>
<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'>e</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'>&#x02211;</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>

This completes the induction that computes 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>e</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow></math>


and 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<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>
</mrow></math>

.






<br/>
<br/>
<b><a name="Standard Math Functions.Cases that Apply Recursion Above" id="Standard Math Functions.Cases that Apply Recursion Above">Cases that Apply Recursion Above</a></b>

<table><tr><td align='left'  valign='top'>

<a href="expforward.xml" target="_top">ExpForward</a></td><td>
Exponential Function Forward Taylor Polynomial Theory</td></tr>
<tr><td>

<a href="logforward.xml" target="_top">LogForward</a></td><td>
Logarithm Function Forward Taylor Polynomial Theory</td></tr>
<tr><td>

<a href="sqrtforward.xml" target="_top">SqrtForward</a></td><td>
Square Root Function Forward Taylor Polynomial Theory</td></tr>
<tr><td>

<a href="sincosforward.xml" target="_top">SinCosForward</a></td><td>
Trigonometric and Hyperbolic Sine and Cosine Forward Theory</td></tr>
<tr><td>

<a href="atanforward.xml" target="_top">AtanForward</a></td><td>
Arctangent Function Forward Taylor Polynomial Theory</td></tr>
<tr><td>

<a href="asinforward.xml" target="_top">AsinForward</a></td><td>
Arcsine Function Forward Taylor Polynomial Theory</td></tr>
<tr><td>

<a href="acosforward.xml" target="_top">AcosForward</a></td><td>
Arccosine Function Forward Taylor Polynomial Theory</td></tr>
<tr><td>

</td></tr>
</table>
<br/>
<b><a name="Standard Math Functions.Special Cases" id="Standard Math Functions.Special Cases">Special Cases</a></b>

<table><tr><td align='left'  valign='top'>

<a href="tan_forward.xml" target="_top">tan_forward</a></td><td>
Tangent and Hyperbolic Tangent Forward Taylor Polynomial Theory</td></tr>
<tr><td>

</td></tr>
</table>

<hr/>Input File: omh/theory/forward_theory.omh

</body>
</html>
