Appendix/SpecialFunctions

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LSUkey.pngSpecial Functions

Gamma Function

Gamma Function

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Resulting Equation

EQ_Gamma01

LSUkey.png

for example, if

 

DLMF §5.5(ii)

Valid for:

   

Complete Elliptic Integrals

Complete Elliptic Integral …

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Resulting Equation

EQ_EllipticIntegral01

… of the First Kind

LSUkey.png

Gradshteyn & Ryzhik (1965), §8.113.1

EQ_EllipticIntegral03

… of the First Kind (alternate expression)

LSUkey.png

Gradshteyn & Ryzhik (1965), §8.113.3

where:  

EQ_EllipticIntegral02

… of the Second Kind

LSUkey.png

Gradshteyn & Ryzhik (1965), §8.114.1

EQ_EllipticIntegral04

… of the Second Kind (alternate expression)

LSUkey.png

Gradshteyn & Ryzhik (1965), §8.114.3

where:  

See also:

Toroidal Function Evaluations

Analytic Expressions & Plots

Toroidal Function Evaluations

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Graphical Representation
(see:  generic caption)

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Resulting Equation

EQ_PminusHalf01

LSUkey.png

      for example …

Abramowitz & Stegun (1995), p. 337, eq. (8.13.1)

Abramowitz & Stegun (1995), p. 337, eq. (8.13.2)

NOTE: We have explicitly demonstrated that an alternate, equivalent expression is:

      where:    

P0minus1Half

EQ_QminusHalf01

LSUkey.png

      for example …

Abramowitz & Stegun (1995), p. 337, eq. (8.13.3)

Abramowitz & Stegun (1995), p. 337, eq. (8.13.4)

Q0minusHalf

EQ_PplusHalf01

LSUkey.png

      for example …

Abramowitz & Stegun (1995), p. 337, eq. (8.13.5)

Abramowitz & Stegun (1995), p. 337, eq. (8.13.6)

NOTE: It appears as though an alternate, equivalent expression is:

      where:    

P0plusHalf

EQ_QplusHalf01

LSUkey.png

Abramowitz & Stegun (1995), p. 337, eq. (8.13.7)

Q0plusHalf

EQ_Q1minusHalf01

LSUkey.png

where:  

(see our associated derivation)

ABSQ1minusHalf

EQ_Q2minusHalf01

LSUkey.png

where:  

(see our associated derivation)

Q2minusHalf

Caption for Plots

Caption for Plots:   Here we explain how we assembled the various plots — shown 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 related discussion) having varying half-integer degrees , , , , and (in association with a separate related discussion) having varying order , .


For each choice of the integer indexes, and , the relevant plot shows how the function, , varies with . (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 [MF53];
  • 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, and , where the relevant values of the elliptic integrals have been pulled directly from tabulated values published in pp. 535 - 537 of [CRC]. (See an accompanying sample of elliptic integral values extracted from [CRC].)
  • 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 Numerical Recipes.


NOTE: The tabulated values of the function, , that appear in Table IX (p. 1923) of [MF53] — also see 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 accompanying discussion — we have plotted the absolute value of the function, .


ADDITIONAL NOTE:   In Example 4 on p. 340 of Abramowitz & Stegun (1995), we can pull one additional data point for comparison; specifically, they provide a high-precision evaluation of . As can be seen in the table of function values immediately below, this is entirely consistent with the lower-precision value that we have extracted from [MF53], and exactly matches the double-precision value we have calculated based on the Numerical Recipes algorithm.

Example Recurrence Relations

The above Toroidal Function Evaluations table provides analytic expressions for the pair of foundation functions, and , and the associated pair of foundation functions, and . 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,"

LSUkey.png

Abramowitz & Stegun (1995), p. 334, eq. (8.5.3)

NOTE: , as well as , satisfies this same recurrence relation.

Specifically, letting and , for all , we have,

    and,

As examples, these two relations have been used to generate columns of numbers in the comparison table shown below for, respectively, the toroidal functions, and . For order-1 and order-2 toroidal functions, the above table provides analytic expressions only for (the functions of the lowest half-integer degree) and . But, as we have detailed in an accompanying discussion, additional order-1 and order-2 expressions can be straightforwardly derived by drawing upon another key recurrence relation, namely,

LSUkey.png

Abramowitz & Stegun (1995), p. 333, eq. (8.5.1)

NOTE: , as well as , satisfies this same recurrence relation.

Specifically, after adopting the association, , we have, when ,

    …    

for

and, when ,

    …    

for

As an example, the first of these two relations has been used to generate a column of numbers in the comparison table shown below for the toroidal function, .

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) , and ; (Bottom half of table) , , , , and . Each function has been evaluated for approximately 23 different argument values in the range, , 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 [MF53]; (Right column) Our double-precision evaluation based on a set of Numerical Recipes algorithms. One exception: The value listed under the "MF53" column for the evaluation of is the high-precision value published on p. 340 of Abramowitz & Stegun (1995); notice that our high-precision evaluation matches all ten digits of their published value.


Top half of Table IX (p. 1923) of [MF53]

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	
Bottom half of Table IX (p. 1923) of [MF53]

ATTENTION:   Widen your browser window, or "zoom out," in order to obtain a proper view of the space-delimited columns of numbers in this table.

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

Relationships Between Various Associated Legendre Functions

Relationships Between Various Associated Legendre Functions

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Resulting Equation

EQ_Toroidal00

LSUkey.png

A. Erdélyi (1953):  Volume I, §3.3.1, p. 140, eq. (7)

 

EQ_Toroidal01

LSUkey.png

A. Erdélyi (1953):  Volume I, §3.11, p. 169, eq. (4)

Valid for:    

  real

       

       

       

  real

EQ_Toroidal02

LSUkey.png

Gil, Segura, & Temme (2000):  eq. (8)

where:    

EQ_Toroidal03

LSUkey.png

A. Erdélyi (1953):  Volume I, §3.7, p. 156, eq. (10)

Valid for:    

 

    and    

 

EQ_Toroidal04

LSUkey.png

Abramowitz & Stegun (1995), p. 334, eq. (8.5.3)

NOTE: , as well as , satisfies this same recurrence relation.

EQ_Toroidal05

LSUkey.png

A. Erdélyi (1953):  Volume I, §3.12, p. 169, eq. (1)

where, and denote any solutions of Legendre's differential equation

EQ_Toroidal06

LSUkey.png

Abramowitz & Stegun (1995), p. 335, eq. (8.9.2)

EQ_Toroidal07

LSUkey.png

Abramowitz & Stegun (1995), p. 333, eq. (8.5.1)

NOTE: , as well as , satisfies this same recurrence relation.

EQ_Toroidal08

LSUkey.png

A. Erdélyi (1953):  Volume I, §3.10, p. 166, eq. (3)

Valid for:    


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