Editing Appendix/SpecialFunctions (section)

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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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