Editing Appendix/SpecialFunctions (section)

===Toroidal Function Evaluations===

====Analytic Expressions &amp; Plots====
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<table border=3 cellpadding=5 cellspacing=1 width="95%" bordercolor="darkblue">
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<font size="+1" color="darkblue">Toroidal Function Evaluations</font>
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To insert a given equation into any Wiki document, type ...<br /><center>
&#123;&#123; Math/<i><font color="red">Template_Name</font></i> &#125;&#125;</center>
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<br />&nbsp;<br />&nbsp;<br />
<font color="red">Graphical Representation</font> <br />(see: &nbsp;[[Appendix/Mathematics/ToroidalFunctions#Caption|generic caption]])
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<font color="red">Template_Name</font>
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<font color="red">Resulting Equation</font>
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[[Template:Math/EQ_PminusHalf01|EQ_PminusHalf01]]
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{{ Math/EQ_PminusHalf01 }}

NOTE:  We have [[Apps/Wong1973Potential#Attempt_.231|explicitly demonstrated]] that an alternate, equivalent expression is:
<table border="0" cellpadding="5" align="center">

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<math>P_{-\frac{1}{2}}(\cosh\eta)</math>
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<math>=</math>
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<math>\frac{\sqrt{2}}{\pi} (\sinh\eta)^{-1 / 2} k K(k)</math>
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<td align="center">&nbsp; &nbsp; &nbsp; where: &nbsp; &nbsp; </td>
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<math>k</math>
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<math>\equiv</math>
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<math>[2/(\coth\eta + 1)]^{1 / 2} \, .</math>
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[[File:P0minus1Half3.png|200px|center|P0minus1Half]]
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[[Template:Math/EQ_QminusHalf01|EQ_QminusHalf01]]
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{{ Math/EQ_QminusHalf01 }}
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[[File:Q0minus1Half3.png|200px|center|Q0minusHalf]]
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[[Template:Math/EQ_PplusHalf01|EQ_PplusHalf01]]
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{{ Math/EQ_PplusHalf01 }}

NOTE:  It appears as though an alternate, equivalent expression is:
<table border="0" cellpadding="5" align="center">

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<math>P_{+\frac{1}{2}}(\cosh\eta)</math>
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<math>=</math>
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<math>\frac{\sqrt{2}}{\pi} (\sinh\eta)^{+1 / 2} k^{-1} E(k)</math>
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<td align="center">&nbsp; &nbsp; &nbsp; where: &nbsp; &nbsp; </td>
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<math>k</math>
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<math>\equiv</math>
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<math>[2/(\coth\eta + 1)]^{1 / 2} \, .</math>
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[[File:P0plus1Half4.png|200px|center|P0plusHalf]]
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[[Template:Math/EQ_QplusHalf01|EQ_QplusHalf01]]
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{{ Math/EQ_QplusHalf01 }}
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[[File:Q0plus1Half3.png|200px|center|Q0plusHalf]]
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[[Template:Math/EQ_Q1minusHalf01|EQ_Q1minusHalf01]]
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{{ Math/EQ_Q1minusHalf01 }}
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[[File:ABSQ1minus1Half3.png|200px|center|ABSQ1minusHalf]]
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[[Template:Math/EQ_Q2minusHalf01|EQ_Q2minusHalf01]]
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{{ Math/EQ_Q2minusHalf01 }}
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[[File:Q2minus1Half3.png|200px|center|Q2minusHalf]]
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====Caption for Plots====
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'''Caption for Plots:''' &nbsp;  Here we explain how we assembled the various plots &#8212; shown [[#Toroidal_Function_Evaluations|immediately above]] in the right-hand column of the "Toroidal Function Evaluations" table  &#8212; that depict the behavior of various associated Legendre (toroidal) functions (see the [[Appendix/Mathematics/ToroidalFunctions#Summary_of_Toroidal_Coordinates_and_Toroidal_Functions|related discussion]]) having varying half-integer degrees <math>P^0_{-\frac{1}{2}}</math>, <math>P^0_{+\frac{1}{2}}</math>, <math>Q^0_{-\frac{1}{2}}</math>, <math>Q^0_{+\frac{1}{2}}</math>, <math>Q^0_{+\frac{3}{2}} \, ,</math> and (in association with a [[Appendix/Mathematics/ToroidalSynopsis01#Q1Q2Summary|separate related discussion]]) having varying order <math>Q^1_{-\frac{1}{2}}</math>,  <math>Q^2_{-\frac{1}{2}}</math>.


For each choice of the integer indexes, <math>n \ge 0</math> and <math>m \ge 0</math>, the relevant plot shows how the function, <math>X^n_{m-\frac{1}{2}}(z)</math>, varies with <math>z</math>.  (Click on the small plot image to view an enlarged image.)  In each plot &hellip;
* The solid green circular markers identify data that has been pulled directly from Table IX (p. 1923) of [<b>[[Appendix/References#MF53|<font color="red">MF53</font>]]</b>];  
* The solid orange circular markers identify function values that we have calculated using the relevant formulae as expressed herein in terms of the complete elliptic integrals, <math>K(k)</math> and <math>E(k)</math>, where the relevant values of the elliptic integrals have been pulled directly from tabulated values published in pp. 535 - 537 of [<b>[[Appendix/References#CRC|<font color="red">CRC</font>]]</b>].  (See an accompanying sample of [[2DStructure/ToroidalCoordinateIntegrationLimits#Evaluation_of_Elliptic_Integrals|elliptic integral values extracted]] from [<b>[[Appendix/References#CRC|<font color="red">CRC</font>]]</b>].)  
* The dashed red curve was also derived using formulae expressed in terms of the complete elliptic integrals, but the ''values'' of the elliptic integrals have been calculated using (double-precision versions of) algorithms drawn from [https://www.amazon.com/Numerical-Recipes-Fortran-Scientific-Computing/dp/052143064X  ''Numerical Recipes''].


NOTE:  The tabulated values of the function, <math>Q^1_{-\frac{1}{2}}</math>, that appear in Table IX (p. 1923) of [<b>[[Appendix/References#MF53|<font color="red">MF53</font>]]</b>] &#8212; also see [[#Comparison_with_Table_IX_from_MF53|immediately below]] &#8212; are all positive, whereas, according to our derivation, they should all be negative.  Therefore, for comparison purposes of this ''specific'' function &#8212; both here and in our [[Appendix/Mathematics/ToroidalSynopsis01#Q1Q2Summary|accompanying discussion]] &#8212; we have plotted the absolute value of the function, <math>|Q^1_{-\frac{1}{2}}(z)|</math>.


ADDITIONAL NOTE: &nbsp; In ''Example 4'' on p. 340 of [https://books.google.com/books?id=MtU8uP7XMvoC&printsec=frontcover&dq=Abramowitz+and+stegun&hl=en&sa=X&ved=0ahUKEwialra5xNbaAhWKna0KHcLAASAQ6AEILDAA#v=onepage&q=Abramowitz%20and%20stegun&f=false Abramowitz &amp; Stegun (1995)], we can pull one additional data point for comparison; specifically, they provide a high-precision evaluation of <math>~Q^0_{-\frac{1}{2}}(z = 2.6) = 1.419337751</math>.  As can be seen in the [[#Comparison_with_Table_IX_from_MF53|table of function values immediately below]], this is entirely consistent with the lower-precision value that we have extracted from [<b>[[Appendix/References#MF53|<font color="red">MF53</font>]]</b>], and exactly matches the double-precision value we have calculated based on the [https://www.amazon.com/Numerical-Recipes-Fortran-Scientific-Computing/dp/052143064X ''Numerical Recipes''] algorithm.
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====Example Recurrence Relations====

The above [[#Analytic_Expressions_.26_Plots|''Toroidal Function Evaluations'']] table provides analytic expressions for the pair of foundation functions, <math>P^0_{-\frac{1}{2}}(z)</math> and <math>P^0_{+\frac{1}{2}}(z)</math>, and the associated pair of foundation functions, <math>Q^0_{-\frac{1}{2}}(z)</math> and <math>Q^0_{+\frac{1}{2}}(z)</math>.  From either pair of foundation functions, expressions for all other zero-order, half-integer degree toroidal functions can be obtained using a relatively simple recurrence relation drawn from the "Key Equation,"

{{ Math/EQ_Toroidal04 }}

Specifically, letting <math>\mu \rightarrow 0</math> and <math>\nu \rightarrow (m - \tfrac{1}{2})</math>, for all <math>~m \ge 2</math>, we have,
<table border="0" cellpadding="5" align="center">

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<math>~P0_{m-\frac{1}{2}}(z)</math>
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<math>=</math>
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<math>4 \biggl[ \frac{m-1}{2m-1} \biggr] z P^0_{m-\frac{3}{2}}(z) - \biggl[ \frac{2m-3}{2m-1}\biggr]P^0_{m-\frac{5}{2}}(z) \, ;</math> &nbsp; &nbsp; and,
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<math>Q^0_{m-\frac{1}{2}}(z)</math>
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<math>=</math>
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<math>4 \biggl[ \frac{m-1}{2m-1} \biggr] z Q^0_{m-\frac{3}{2}}(z) - \biggl[ \frac{2m-3}{2m-1}\biggr]Q^0_{m-\frac{5}{2}}(z) \, .</math>
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As examples, these two relations have been used to generate columns of numbers in the [[#Comparison_with_Table_IX_from_MF53|comparison table shown below]] for, respectively, the toroidal functions, <math>P^0_{+\frac{3}{2}}(z)</math> and <math>Q^0_{+\frac{3}{2}}(z)</math>. For order-1 and order-2 toroidal functions, the above table provides analytic expressions only for (the functions of the lowest half-integer degree) <math>Q^1_{-\frac{1}{2}}(z)</math> and  <math>Q^2_{-\frac{1}{2}}(z)</math>.  But, as we have detailed in an [[Appendix/Mathematics/ToroidalSynopsis01#Evaluating_Q2.CE.BD|accompanying discussion]], additional order-1 and order-2 expressions can be straightforwardly derived by drawing upon another key recurrence relation, namely,

{{ Math/EQ_Toroidal07 }}

Specifically, after adopting the association, <math>\nu \rightarrow (n - \tfrac{1}{2})</math>, we have, when <math>\mu = 0</math>,

<table border="0" cellpadding="5" align="center">

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<math>Q_{n - \frac{1}{2}}^{1}(z)</math>
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<math>=</math>
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<math>
(n-\tfrac{1}{2}) (z^2-1)^{-\frac{1}{2}} [z Q_{n - \frac{1}{2}}(z) - Q_{n - \frac{3}{2}}(z)]
</math>
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  <td allign="center">&nbsp; &nbsp; &hellip; &nbsp; &nbsp;</td>
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for <math>n \ge 1 \, ,</math>
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and, when <math>~\mu = 1</math>,
<table border="0" cellpadding="5" align="center">

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<math>Q_{n - \frac{1}{2}}^{2}(z)</math>
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<math>=</math>
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<math>
(z^2-1)^{-\frac{1}{2}} \{ (n-\tfrac{3}{2}) z Q^1_{n - \frac{1}{2}}(z) - (n+\tfrac{1}{2})Q^1_{n - \frac{3}{2}}(z)\}
</math>
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  <td allign="center">&nbsp; &nbsp; &hellip; &nbsp; &nbsp;</td>
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for <math>n \ge 1 \, .</math>
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As an example, the first of these two relations has been used to generate a column of numbers in the [[#Comparison_with_Table_IX_from_MF53|comparison table shown below]] for the toroidal function, <math>Q^1_{+\frac{1}{2}}(z)</math>.

====Comparison with Table IX from MF53====

To facilitate ''copying &amp; pasting'' for immediate use by other researchers, here we present in a tab-delimited, plain-text format the evaluation of nine separate toroidal functions:  (''Top half of table'') <math>~P^0_{-\frac{1}{2}}</math>, <math>~P^0_{+\frac{1}{2}}</math> and <math>~P^0_{+\frac{3}{2}}</math>; (''Bottom half of table'') <math>~Q^0_{-\frac{1}{2}}</math>, <math>~Q^1_{-\frac{1}{2}}</math>,  <math>~Q^2_{-\frac{1}{2}}</math>, <math>~Q^0_{+\frac{1}{2}}</math>, <math>~Q^1_{+\frac{1}{2}}</math> and <math>~Q^0_{+\frac{3}{2}}</math>.  Each function has been evaluated for approximately 23 different argument values in the range, <math>~1.0 \le z \le 9.0</math>, and, for each function, two columns of function values have been presented:  (''Left column'') Low-precision evaluation extracted directly from Table IX (p. 1923) of [<b>[[User:Tohline/Appendix/References#MF53|<font color="red">MF53</font>]]</b>]; (''Right column'') Our double-precision evaluation based on a set of [https://www.amazon.com/Numerical-Recipes-Fortran-Scientific-Computing/dp/052143064X ''Numerical Recipes''] algorithms.  One exception:  The value listed under the "MF53" column for the evaluation of <math>~Q^0_{-\frac{1}{2}}(z=2.6)</math> is the high-precision value published on p. 340 of [https://books.google.com/books?id=MtU8uP7XMvoC&printsec=frontcover&dq=Abramowitz+and+stegun&hl=en&sa=X&ved=0ahUKEwialra5xNbaAhWKna0KHcLAASAQ6AEILDAA#v=onepage&q=Abramowitz%20and%20stegun&f=false Abramowitz &amp; Stegun (1995)]; notice that our high-precision evaluation matches all ten digits of their published value.


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Top half of Table IX (p. 1923) of [<b>[[Appendix/References#MF53|<font color="red">MF53</font>]]</b>]
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<pre>
z	    P0m1Half(z)		   P0p1Half(z)		   P0p3Half(z)	
	 MF53	  Our Calc.	 MF53	  Our Calc.	 MF53	  Our Calc.	
1.0	1.0000			1.0000			1.0000		
1.2	0.9763	9.763155118E-01	1.0728	1.072784040E+00	1.3910	1.391015961E+00	
1.4	0.9549	9.549467781E-01	1.1416	1.141585331E+00	1.8126	1.812643692E+00	
1.6	0.9355	9.355074856E-01	1.2070	1.206963827E+00	2.2630	2.263020336E+00	
1.8	0.9177	9.176991005E-01	1.2694	1.269362428E+00	2.7406	2.740570128E+00	
2.0	0.9013	9.012862994E-01	1.3291	1.329138155E+00	3.2439	3.243939648E+00	
2.2	0.8861	8.860804115E-01	1.3866	1.386583505E+00	3.7719	3.771951476E+00	
2.4	0.8719	8.719279330E-01	1.4419	1.441941436E+00	4.3236	4.323569952E+00	
2.6	0.8587	8.587023595E-01	1.4954	1.495416274E+00	4.8979	4.897875630E+00	
2.8	0.8463	8.462982520E-01	1.5472	1.547181667E+00	5.4941	5.494045473E+00	
3.0	0.8346	8.346268417E-01	1.5974	1.597386605E+00	6.1113	6.111337473E+00	
3.5	0.8082	8.081851582E-01	1.7169	1.716877977E+00	7.7427	7.742702172E+00	
4.0	0.7850	7.849616703E-01	1.8290	1.828992729E+00	9.4930	9.492973996E+00	
4.5	0.7643	7.643076802E-01	1.9349	1.934919997E+00	11.3555	1.135475076E+01	
5.0	0.7457	7.457491873E-01	2.0356	2.035563839E+00	13.3220	1.332184253E+01	
5.5	0.7289	7.289297782E-01	2.1316	2.131629923E+00	15.3890	1.538897617E+01	
6.0	0.7136	7.135750093E-01	2.2237	2.223681177E+00	17.5520	1.755159108E+01	
6.5	0.6995	6.994692725E-01	2.3122	2.312174942E+00	19.8060	1.980569307E+01	
7.0	0.6864	6.864402503E-01	2.3975	2.397488600E+00	22.1480	2.214774685E+01	
7.5	0.6743	6.743481630E-01	2.4799	2.479937758E+00	24.5750	2.457459486E+01	
8.0	0.6631	6.630781433E-01	2.5598	2.559789460E+00	27.0830	2.708339486E+01	
8.5		6.525347093E-01		2.637271986E+00		2.967157094E+01	
9.0		6.426376817E-01		2.712582261E+00		3.233677457E+01	
</pre>
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<div align="center">Bottom half of Table IX (p. 1923) of [<b>[[Appendix/References#MF53|<font color="red">MF53</font>]]</b>]</div>
<font color="red">ATTENTION:</font> &nbsp; Widen your browser window, or "zoom out," in order to obtain a proper view of the space-delimited columns of numbers in this table.
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<pre>
z	    Q0m1Half(z)		    Q1m1Half(z)			    Q2m1Half(z)		    Q0p1Half(z)		    Q1p1Half(z)			    Q0p3Half(z)
	 MF53	   Our Cal.	 MF53	   Our Calc.		 MF53	   Our Calc.	 M53	   Our Calc.	 MF53	   Our Calc.		 MF53	   Our Calc.
1.1 	2.8612	2.861192872E+00	2.3661	-2.366084077E+00	10.6440	1.064378304E+01	0.9788	9.787602829E-01	1.9471	-1.947110839E+00	0.4818	4.817841242E-01
1.2 	2.5010	2.500956508E+00	1.7349	-1.734890983E+00	5.6518	5.651832631E+00	0.6996	6.995548314E-01	1.2524	-1.252395745E+00	0.2856	2.856355610E-01
1.4 	2.1366	2.136571733E+00	1.2918	-1.291802851E+00	3.1575	3.157491205E+00	0.4598	4.597941602E-01	0.7618	-7.618218821E-01	0.14609	1.460918547E-01
1.6 	1.9229	1.922920866E+00	1.0943	-1.094337965E+00	2.3230	2.323018870E+00	0.3430	3.430180260E-01	0.5501	-5.500770475E-01	0.09080	9.079816684E-02
1.8 	1.7723	1.772268479E+00	0.9748	-9.748497733E-01	1.9018	1.901788930E+00	0.2720	2.720401772E-01	0.4285	-4.284853031E-01	0.06214	6.214026586E-02
2.0 	1.6566	1.656638170E+00	0.8918	-8.917931374E-01	1.6454	1.645348489E+00	0.2240	2.240142929E-01	0.3489	-3.488955345E-01	0.04516	4.515872426E-02
2.2 	1.5634	1.563378886E+00	0.8293	-8.292825549E-01	1.4712	1.471197798E+00	0.18932	1.893229696E-01	0.29263	-2.926294028E-01	0.03422	3.422108228E-02
2.4     1.4856	1.485653983E+00	0.7798	-7.797558474E-01	1.3441	1.344108936E+00	0.16312	1.631167365E-01	0.25076	-2.507568731E-01	0.02676	2.675556229E-02
2.6 1.419337751	1.419337751E+00	0.7391	-7.390875295E-01	1.2465	1.246521876E+00	0.14266	1.426580119E-01	0.21842	-2.184222751E-01	0.02143	2.143519083E-02
2.8	1.3617	1.361744950E+00	0.7048	-7.048053314E-01	1.1687	1.168702464E+00	0.12628	1.262756033E-01	0.19274	-1.927423405E-01	0.01751	1.751393553E-02
3.0	1.3110	1.311028777E+00	0.6753	-6.753219405E-01	1.1048	1.104816977E+00	0.11289	1.128885424E-01	0.17189	-1.718911443E-01	0.01454	1.454457729E-02
3.5	1.2064	1.206444997E+00	0.6163	-6.163068170E-01	0.9846	9.846190928E-01	0.08824	8.824567577E-02	0.13380	-1.338040913E-01	0.00966	9.664821286E-03
4.0	1.1242	1.124201960E+00	0.5713	-5.712994484E-01	0.8990	8.990205764E-01	0.07154	7.154134054E-02	0.10819	-1.081900595E-01	0.00682	6.819829619E-03
4.5	1.0572	1.057164923E+00	0.5353	-5.353494651E-01	0.8339	8.338659751E-01	0.05957	5.956966068E-02	0.08993	-8.992645608E-02	0.00503	5.029656514E-03
5.0	1.0011	1.001077380E+00	0.5057	-5.056928088E-01	0.7820	7.819717783E-01	0.05063	5.062950976E-02	0.07634	-7.633526879E-02	0.00384	3.837604899E-03
5.5	0.9532	9.532056775E-01	0.4806	-4.806378723E-01	0.7393	7.392682950E-01	0.04374	4.373774515E-02	0.06588	-6.588433822E-02	0.00301	3.008238619E-03
6.0	0.9117	9.116962715E-01	0.4591	-4.590784065E-01	0.7033	7.032568965E-01	0.03829	3.828867029E-02	0.05764	-5.763649873E-02	0.00241	2.410605139E-03
6.5	0.87524	8.752387206E-01	0.44025	-4.402537373E-01	0.67231	6.723067009E-01	0.03389	3.389003482E-02	0.05099	-5.098806037E-02	0.00197	1.967394932E-03
7.0	0.84288	8.428751774E-01	0.42362	-4.236198508E-01	0.64530	6.453008278E-01	0.03028	3.027740449E-02	0.04553	-4.553369214E-02	0.00163	1.630716095E-03
7.5	0.81389	8.138862008E-01	0.40877	-4.087751846E-01	0.62144	6.214442864E-01	0.02727	2.726650960E-02	0.04099	-4.099183107E-02	0.00137	1.369695722E-03
8.0	0.78772	7.877190099E-01	0.39542	-3.954155185E-01	0.60015	6.001530105E-01	0.02473	2.472532098E-02	0.03716	-3.716124286E-02	0.00116	1.163753807E-03
8.5		7.639406230E-01		-3.833053056E-01		5.809864341E-01		2.255696890E-02		-3.389458114E-02		9.987731857E-04
9.0		7.422062367E-01		-3.722587645E-01		5.636047532E-01		2.068890884E-02		-3.108168349E-02		8.648271474E-04
</pre>
</td>
</tr>
</table>
</div>