Appendix/SpecialFunctions - Revision history

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Gamma Function

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=[[File:LSUkey.png|50px]]Special Functions=

===Gamma Function===

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<font size="+1" color="darkblue">Gamma Function</font>
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<br /> <br /> <br />
<font color="red">See also …</font>
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[[Template:Math/EQ_Gamma01|EQ_Gamma01]]
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{{ Math/EQ_Gamma01 }}
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[https://authors.library.caltech.edu/43491/1/Volume%201.pdf A. Erdélyi (1953)]:  Volume I, §1.2, p. 3, eq. (6)
[https://en.wikipedia.org/wiki/Gamma_function#General Wikipedia]
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===Complete Elliptic Integrals===

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<font size="+1" color="darkblue">Complete Elliptic Integral …</font>
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<br /> <br /> <br />
<font color="red">See also …</font>
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[[Template:Math/EQ_EllipticIntegral01|EQ_EllipticIntegral01]]
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<font size="+1" color="darkblue">… of the First Kind</font><br />
{{ Math/EQ_EllipticIntegral01 }}
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<td colspan=1 align="left">
* [https://dlmf.nist.gov/19.5.E1 DLMF §19.5.1] <br />
* [https://mathworld.wolfram.com/CompleteEllipticIntegraloftheFirstKind.html Wolfram's Mathworld]<br />
* [https://en.wikipedia.org/wiki/Elliptic_integral#Complete_elliptic_integral_of_the_first_kind Wikipedia]<br />
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[[Template:Math/EQ_EllipticIntegral03|EQ_EllipticIntegral03]]
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<font size="+1" color="darkblue">… of the First Kind</font><font color="darkblue"> (alternate expression)</font><br />
{{ Math/EQ_EllipticIntegral03 }}
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*
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[[Template:Math/EQ_EllipticIntegral02|EQ_EllipticIntegral02]]
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<font size="+1" color="darkblue">… of the Second Kind</font><br />
{{ Math/EQ_EllipticIntegral02 }}
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* [https://dlmf.nist.gov/19.5.E2 DLMF §19.5.2] <br />
* [https://mathworld.wolfram.com/CompleteEllipticIntegraloftheSecondKind.html Wolfram's MathWorld]<br />
* [https://en.wikipedia.org/wiki/Elliptic_integral#Complete_elliptic_integral_of_the_second_kind Wikipedia]
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[[Template:Math/EQ_EllipticIntegral04|EQ_EllipticIntegral04]]
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<font size="+1" color="darkblue">… of the Second Kind</font><font color="darkblue"> (alternate expression)</font><br />
{{ Math/EQ_EllipticIntegral04 }}
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*
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See also:
* [https://www-jstor-org.libezp.lib.lsu.edu/stable/2004103?seq=1#metadata_info_tab_contents W. J. Cody (1965, Mathematics of Computation, Vol. 19, No. 89, pp. 105 - 112)], "<i>Chebyshev Approximations for the Complete Elliptic Integrals K and E</i>".
* "[https://www.ams.org/journals/mcom/1965-19-090/S0025-5718-1965-0178563-0/S0025-5718-1965-0178563-0.pdf Chebyshev Polynomial Expansions of Complete Elliptic Integrals]," by W. J. Cody (Argonne National Laboratory)

===Toroidal Function Evaluations===

====Analytic Expressions & Plots====
<div align="center">
<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>
{{ Math/<i><font color="red">Template_Name</font></i> }}</center>
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<br /> <br /> <br />
<font color="red">Graphical Representation</font> <br />(see:  [[Appendix/Mathematics/ToroidalFunctions#Caption|generic caption]])
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<font color="red">Template_Name</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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<td align="right">
<math>P_{-\frac{1}{2}}(\cosh\eta)</math>
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<math>=</math>
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<td align="left">
<math>\frac{\sqrt{2}}{\pi} (\sinh\eta)^{-1 / 2} k K(k)</math>
</td>
<td align="center">      where:     </td>
<td align="right">
<math>k</math>
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<td align="center">
<math>\equiv</math>
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<math>[2/(\coth\eta + 1)]^{1 / 2} \, .</math>
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</table>

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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">

<tr>
<td align="right">
<math>P_{+\frac{1}{2}}(\cosh\eta)</math>
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<td align="center">
<math>=</math>
</td>
<td align="left">
<math>\frac{\sqrt{2}}{\pi} (\sinh\eta)^{+1 / 2} k^{-1} E(k)</math>
</td>
<td align="center">      where:     </td>
<td align="right">
<math>k</math>
</td>
<td align="center">
<math>\equiv</math>
</td>
<td align="left">
<math>[2/(\coth\eta + 1)]^{1 / 2} \, .</math>
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</table>
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<td colspan=1 align="left">
[[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====
<div align="center" id="Caption">
<table border="1" cellpadding="8" width="95%" align="center">
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'''Caption for Plots:'''   Here we explain how we assembled the various plots — shown [[#Toroidal_Function_Evaluations|immediately above]] in the right-hand column of the "Toroidal Function Evaluations" table — 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 …
* 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>] — also see [[#Comparison_with_Table_IX_from_MF53|immediately below]] — are all positive, whereas, according to our derivation, they should all be negative. Therefore, for comparison purposes of this ''specific'' function — both here and in our [[Appendix/Mathematics/ToroidalSynopsis01#Q1Q2Summary|accompanying discussion]] — we have plotted the absolute value of the function, <math>|Q^1_{-\frac{1}{2}}(z)|</math>.

ADDITIONAL NOTE:   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 & 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">

<tr>
<td align="right">
<math>~P0_{m-\frac{1}{2}}(z)</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<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>     and,
</td>
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<tr>
<td align="right">
<math>Q^0_{m-\frac{1}{2}}(z)</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<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>
</td>
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</table>

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">

<tr>
<td align="right">
<math>Q_{n - \frac{1}{2}}^{1}(z)</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<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>
</td>
<td allign="center">    …    </td>
<td align="left">
for <math>n \ge 1 \, ,</math>
</td>
</tr>
</table>

and, when <math>~\mu = 1</math>,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>Q_{n - \frac{1}{2}}^{2}(z)</math>
</td>
<td align="center">
<math>=</math>
</td>
<td align="left">
<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>
</td>
<td allign="center">    …    </td>
<td align="left">
for <math>n \ge 1 \, .</math>
</td>
</tr>
</table>
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 & 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 & Stegun (1995)]; notice that our high-precision evaluation matches all ten digits of their published value.

<div align="center" id="TabulatedValues">
<table border="1" cellpadding="8" width="90%" align="center">
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<td align="center">
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>   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>
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===Relationships Between Various Associated Legendre Functions===

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<font size="+1" color="darkblue">Relationships Between Various Associated Legendre Functions</font>
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To insert a given equation into any Wiki document, type ...<br /><center>
{{ Math/<i><font color="red">Template_Name</font></i> }}</center>
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<br /> <br /> <br />
<font color="red">See also …</font>
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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_Toroidal00|EQ_Toroidal00]]
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{{ Math/EQ_Toroidal00 }}
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[[Template:Math/EQ_Toroidal01|EQ_Toroidal01]]
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* [http://adsabs.harvard.edu/abs/1940QJMat..11..222C T. G. Cowling (1940)]:   p. 223 (note sign discrepancy in argument of <math>Q_\nu</math>)
* [https://dlmf.nist.gov/14.18.E5 DLMF §14.18.5]
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[[Template:Math/EQ_Toroidal02|EQ_Toroidal02]]
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* [http://adsabs.harvard.edu/abs/2000AN....321..363C Cohl et al. (2000)], eq. (34)
* [https://dlmf.nist.gov/14.19#v DLMF §14.19.v] together with [https://dlmf.nist.gov/14.3.E10 DLMF §14.3.10]
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[[Template:Math/EQ_Toroidal03|EQ_Toroidal03]]
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[[Template:Math/EQ_Toroidal04|EQ_Toroidal04]]
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* [http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl & Tohline (1999)], §2.2.2, eq. (25)<br />
* [https://dl-acm-org.libezp.lib.lsu.edu/citation.cfm?id=365474&picked=prox Guatschi (1965)], p. 490, '''procedure''' ''toroidal
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[[Template:Math/EQ_Toroidal05|EQ_Toroidal05]]
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*
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[[Template:Math/EQ_Toroidal06|EQ_Toroidal06]]
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* [https://authors.library.caltech.edu/43491/1/Volume%201.pdf Erdélyi (1953)]:  Volume I, §3.8, p. 162, eq. (21)
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[[Template:Math/EQ_Toroidal07|EQ_Toroidal07]]
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*
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[[Template:Math/EQ_Toroidal08|EQ_Toroidal08]]
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*
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