Mathematica examples relevant to Legendre functions

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Bertram George
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1 Mathematica eamples relevant to Legendre functions Legendre Polynomials are built in Here is Legendre s equation, and Mathematica recognizes as being solved by Legendre polynomials (LegendreP) and the second solution (LegendreQ) DSolve@H ^ L y ''@D y '@D + n Hn + L y@d, y@d, D y@d C@D LegendreP@n, D + C@D LegendreQ@n, D<< Evaluating and plotting first few Legendre polynomials Table@n, LegendreP@n, D<, n,, 6<D TableForm 4 6 H + L H + L H + 4 L H L I M legto = Table@LegendreP@n, D, n,, <D :,, I + M, I + M> Plot of first 4 Legendre Polynomials

2 <, PlotStyle Thick, Red<, Thick, Green<, Thick, Blue<, Thick, Black<<, LabelStyle "Medium", AesLabel, P<, PlotLabel "First 4 Legendre Polynomials"D First 4 Legendre Polynomials P Normalization and orthogonality LegendreP@n, D Integrate@LegendreP@, D LegendreP@, D,,, <D The other Legendre solutions (which diverge at =, ) These are the second solutions to Legendre s equation, usually denoted Q LegendreQ@, D Log@ D + Log@ + D LegendreQ@, D + Log@ D + Log@ + D LegendreQ@, D + I + M Log@ D + Log@ + D LegendreQ@, D I M Log@ D + Log@ + D

3 D, D, D, D<,,, <, PlotStyle Thick, Red<, Thick, Green<, Thick, Blue<, Thick, Black<<, PlotLabel "Second solutions l=,,,"d Second solutions l=,,,.... Checking Generating Function _D := h + h ^ D Series@phi@h, D, h,, 6<D +h+ + h + I M h h + I + 4 M h4 + I M h6 + O@hD7 Compare to see that the coefficient on h^n is indeed P_n Table@SeriesCoefficient@phi@h, D, h,, l<d Simplify, LegendreP@l, D<, l,, 6<D TableForm H + L H + L H + 4 L H L I M H + L H + L H + 4 L H L I M The Simple Eamples of Legendre Polynomials in Physics The a single charged particle s ' l= = S Hr'Ll l+ l= r r potential can be written in terms of Legendre Prolynomials: Pl HcosΓL where cosγ is the angle between (the position vector to the point of observation) and ' (the position vector to the point where the charge is located). This series is only convergent for r>r (away from the source).

4 4 l= ' = S Hr'Ll l+ l= r Pl HcosΓL where cosγ is the angle between (the position vector to the point of observation) and ' (the position vector to the point where the charge is located). This series is only convergent for r>r (away from the source). A simple case is a point charge shifted off of the origin at point (,). Looking at the slice at y=, cosγ=/+ depending if you are on the right or the left of the point charge, in other words cosγ=sign[]. Lastly we can simplify because the Legendre Polynomials are normalized Pl HL = and are even/odd for even/odd l. This means we can replace LegendreP[l,Sign[]] with just (Sign[])^l ManipulateBPlotB: H + L, SumB l+ H Sign@DLl, l,, ll<f>,,, <, PlotRange., <, AspectRatio, PlotLegends :" Ó Ó ' l=ll ", " S l= Hr 'Ll r l+ Pl HcosΓL">, PlotStyle Orange, Blue<F, ll,,, <F ll.. '. l=ll S Hr'Ll l+ l= r Pl HcosΓL. As we add higher moments the approimation becomes better in the region r>r. To see this in D:

5 ManipulateB PlotDB: H + L + y, SumB LegendrePBl, l+ + y +y,, <, y,, <, AspectRatio, PlotRange,.<, PlotStyle Orange, Blue<, PlotLegends :" Ó Ó ' l=ll ", " S l= Hr 'Ll r l+ F, l,, ll<f>, Pl HcosΓL">F, ll,,, <F ll.4 ' l=ll S.. Hr'Ll l+ l= r Pl HcosΓL For a slightly more interesting eample, we can look at a physical dipole, with a positive charge at (,) and a negative charge at (,). Again in a y= slice, adding the Legendre polynomial for each term by term looks like Pl HSign@DL Pl HSign@DL For any even l, this will vanish and for odd l it will equal *Sign[]. This makes sense because the charge distribution is odd along the ais. Considering there is no net charge it makes sense there will be no monopole term.

6 6 ManipulateBPlotB: H + L H L, SumB l+ H * Sign@DL, l,, ll, <F>,,, <, PlotRange, <, AspectRatio, PlotStyle Orange, Blue<, PlotLegends "Eact", "Sum up to ll"<f, ll,,, <F ll 4 Eact 4 Sum up to ll

7 ManipulateBPlotDB: H + L + y H L + y l+ +y +y, SumB F LegendrePBl, LegendrePBl, 7 F, l,, ll, <F>, +y,, <, y,, <, AspectRatio, PlotRange.,.<, PlotStyle Orange, Blue<, PlotLegends "Eact", "Sum up to ll"<f, ll,,, <F ll. Eact Sum up to ll.. To get an idea of how quickly the sum converges, lets look at the Log of the absolute value of the differences, dipoleres@_, y_, ll_d := LogBAbsB H + L + y LegendrePBl, l+ +y +y H L + y F LegendrePBl, SumB +y F, l,, ll, <FFF

8 y, lld,,, <, y,, <, AspectRatio, PlotRange 4, <, PlotLegends Sum up to lldd"<d, ll,,, <D ll Sum up to lldd 4 We see it converges quickly and tends to be a better approimation for larger r. We can also see if we bring the two point charges closer to the origin, the l= dipole term becomes eact. This is the same as if we were zooming out, observing from greater r.

9 ManipulateBPlotB: H + ΕL Ε H ΕL, H Sign@DL>,,, <, PlotRange, <, AspectRatio, PlotStyle Orange, Blue<, PlotLegends "Eact", "Sum up to ll"<f, Ε,, <F Ε Eact Sum up to ll Eample of Legendre series for step function (Boas Sec. 9.) Here are the series coefficients. Step function is implemented by integrating from to. In[]:= c@n_d := Integrate@LegendreP@n, D,,, <D H n + L 9

10 In[]:= n,, 7<D TableForm Out[]//TableForm= 4 In[4]:= In[]:= legsum@_, m_d := Epand@Sum@c@nD LegendreP@n, D, n,, m<dd legsum@_d = legsum@, D + Out[]= In[6]:= 4 legsum7@_d = legsum@, 7D In[7]:= Out[6]= legsum@_d = legsum@, D; Caution! When Mathematica is working with large polynoimials with large coefficents that require eact cancelations, sometimes Mathematica will not give the correct answer due to not working to high enough numerical precision. When working with a decimal, In[]:= Out[]= legsum@.9d. When keeping it as a fraction In[9]:= Out[9]= In[4]:= Out[4]= legsum@9 D % N.994

11 Plot of approimation with up to P_, P_7 and P_ In[4]:= WorkingPrecision, PlotStyle Thick, Blue<, Thick, Green<, Thick, Red<<, PlotRange >, <, LabelStyle "Large", PlotLabel "Legendre series for a step", PlotLegends :" S cl Pl HL", " S cl Pl HL", " S cl Pl HL">F l= l=7 l= l= l= l= Legendre series for a step.... Out[4]= S cl Pl HL l= l= S cl Pl HL l=7 l= S cl Pl HL l= l= Lets play with this a little more (caution, very large values of l may cause Mathematica to slow down as l the lth polynomial has ~ + terms (rounded down)

12 In[4]:= ldd<,,, <, WorkingPrecision, PlotStyle Thick, Blue<, Thick, Green<, Thick, Red<<, PlotRange >, <, PlotLegends " S cl Pl HL"F, l,,, <F l=ll l= l... Out[4]= S cl Pl HL l=ll. l= Integrate::ilim : Invalid integration variable or limithsl in.99999,, <. Integrate::ilim : Invalid integration variable or limithsl in.994,, <. Integrate::ilim : Invalid integration variable or limithsl in.96,, <. General::stop : Further output of Integrate::ilim will be suppressed during this calculation. Associated Legendre polynomials Pm n HL is given by LegendreP[n,m,] m Note that Pn HL is the same as Pn HL, and that Pm n HL and Pn HL are proportional. tabassoc@n_d := Table@n, m, LegendreP@n, m, D<, m, n, n<d tabassoc@d TableForm

13 TableForm H L H + L H + L Plot@LegendreP@,, D, LegendreP@,, D<,,, <, PlotStyle Thick, Blue<, Thick, Green<, Thick, Red<<, PlotLabel "Associated Legendres for n="d Associated Legendres for n= Plot@LegendreP@,, D, LegendreP@,, D, LegendreP@,, D<,,, <, PlotStyle Thick, Blue<, Thick, Green<, Thick, Red<<, PlotLabel "Associated Legendres for n="d Associated Legendres for n=....

Lecture 22 Appendix B (More on Legendre Polynomials and Associated Legendre functions)

Lecture 22 Appendix B (More on Legendre Polynomials and Associated Legendre functions) Lecture Appendix B (More on Legendre Polynomials and Associated Legendre functions) Ex.7: Let's use Mathematica to solve the equation and then verify orthogonality. First In[]:= Out[]= DSolve y'' x n^y