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<center><b><big><big>Tangent and Hyperbolic Tangent Reverse Mode Theory</big></big></b></center>
<br/>
<b><big><a name="Notation" id="Notation">Notation</a></big></b>
<br/>
We use the reverse theory
<a href="reversetheory.xml#Standard Math Functions" target="_top"><span style='white-space: nowrap'>standard&#xA0;math&#xA0;function</span></a>

definition for the functions 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>H</mi>
</mrow></math>

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

.
In addition, we use the forward mode notation in <a href="tan_forward.xml" target="_top"><span style='white-space: nowrap'>tan_forward</span></a>
 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>

, 
<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 
<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>

.

<br/>
<br/>
<b><big><a name="Eliminating Y(t)" id="Eliminating Y(t)">Eliminating Y(t)</a></big></b>
<br/>
For 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">&gt;</mo>
<mn>0</mn>
</mrow></math>

, the forward mode coefficients are given by

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mn>-1</mn>
<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>
<mrow><mi mathvariant='italic'>j</mi>
<mn>-1</mn>
</mrow>
</munderover>
<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'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>k</mi>
<mn>-1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow></math>

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

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

 is the same as 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>G</mi>
</mrow></math>

 
except that 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mn>-1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow></math>

 is replaced as a function of the Taylor 
coefficients 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>

.
To be specific, 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>

,

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>H</mi>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>G</mi>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
<mo stretchy="false">+</mo>
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>G</mi>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mn>-1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mn>-1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
</mtd></mtr><mtr><mtd columnalign="right" >
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>G</mi>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
<mo stretchy="false">+</mo>
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>G</mi>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mn>-1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
<mn>2</mn>
<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>
<mn>-1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mtd></mtr></mtable>
</mrow></math>

<br/>
<b><big><a name="Positive Orders Z(t)" id="Positive Orders Z(t)">Positive Orders Z(t)</a></big></b>
<br/>
For order 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">&gt;</mo>
<mn>0</mn>
</mrow></math>

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

 is the same as 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>G</mi>
</mrow></math>

 except that

<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>
</mrow>
</msup>
</mrow></math>

 is expressed as a function of 
the 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>

, and the
lower order Taylor coefficients for 
<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>

, 
<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>

.

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<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'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">&#x000B1;</mo>
<mfrac><mrow><mn>1</mn>
</mrow>
<mrow><mi mathvariant='italic'>j</mi>
</mrow>
</mfrac>
<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'>x</mi>
<mrow><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'>j</mi>
<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>1</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">&#x02026;</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>j</mi>
</mrow></math>

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

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

 is given by

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>H</mi>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>G</mi>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
<mo stretchy="false">+</mo>
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>G</mi>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
</mtd></mtr><mtr><mtd columnalign="right" >
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>G</mi>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
<mo stretchy="false">+</mo>
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>G</mi>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
<mrow><mo stretchy="true">[</mo><mrow><mi mathvariant='normal'>&#x003B4;</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">&#x000B1;</mo>
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>G</mi>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
<mfrac><mrow><mi mathvariant='italic'>k</mi>
</mrow>
<mrow><mi mathvariant='italic'>j</mi>
</mrow>
</mfrac>
<msup><mi mathvariant='italic'>y</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>
</mrow><mo stretchy="true">]</mo></mrow>
</mtd></mtr></mtable>
</mrow></math>

where 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='normal'>&#x003B4;</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
</mrow></math>

 is one if 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>k</mi>
</mrow></math>

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

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

 with respect to 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>y</mi>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>k</mi>
</mrow>
</msup>
</mrow></math>

,
is given by

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>H</mi>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msup><mi mathvariant='italic'>y</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>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>G</mi>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msup><mi mathvariant='italic'>y</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>
</mrow>
</mfrac>
<mo stretchy="false">+</mo>
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>G</mi>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msup><mi mathvariant='italic'>y</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>
</mrow>
</mfrac>
</mtd></mtr><mtr><mtd columnalign="right" >
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>G</mi>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msup><mi mathvariant='italic'>y</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>
</mrow>
</mfrac>
<mo stretchy="false">&#x000B1;</mo>
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>G</mi>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
<mfrac><mrow><mi mathvariant='italic'>k</mi>
</mrow>
<mrow><mi mathvariant='italic'>j</mi>
</mrow>
</mfrac>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mi mathvariant='italic'>k</mi>
</mrow>
</msup>
</mtd></mtr></mtable>
</mrow></math>

<br/>
<b><big><a name="Order Zero Z(t)" id="Order Zero Z(t)">Order Zero Z(t)</a></big></b>
<br/>
The order zero coefficients for the tangent and hyperbolic tangent are

<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" >
<mrow><mo stretchy="true">{</mo><mrow><mtable rowalign="center" ><mtr><mtd columnalign="center" >
<mi>tan</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="center" >
<mi>tanh</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></mtable>
</mrow><mo stretchy="true"> </mo></mrow>
</mtd></mtr></mtable>
</mrow></math>

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

 is the same as 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>G</mi>
</mrow></math>

 except that

<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>

 is expressed as a function of the 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>

.
In this case,

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>H</mi>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>G</mi>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
<mo stretchy="false">+</mo>
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>G</mi>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
</mtd></mtr><mtr><mtd columnalign="right" >
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>G</mi>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
<mo stretchy="false">+</mo>
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>G</mi>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msup><mi mathvariant='italic'>z</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">&#x000B1;</mo>
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mtd></mtr></mtable>
</mrow></math>


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

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