THE LEGENDRE OLYNOMIALS AND THEIR ROERTIES The gravitatioal potetial ψ at a poit A at istace r from a poit mass locate at B ca be represete by the solutio of the Laplace equatio i spherical cooriates. I its simplest form oe has- ψ Cost. ψ r with solutio ψ A r r r r If oe ow thiks of obtaiig the potetial of a istribute mass, the solutio becomes- Vol ψ A cost. r Vol R cos θ Where R is the istace from the ceter of mass at C to the eteral poit A, R/R < is the ratio of istace BC to AC, a θ is the agle betwee lies AC a BC. As first oe by Legere, oe ca epa the raical i the eomiator of this itegral to get- 35cos θ [ 3cos θ [cos θ ] [ cos θ 3cos θ 3 5 63cos θ ] [ 5 3 5cos θ ] [ 7cos θ 3 3 3cos θ ] 5cos θ ]... Substitutig this epasio ito the above itegral leas to the well kow multipole epasio for the potetial of a gravitatioal mass as viewe from a poit A outsie of this mass. For us the importat poit here are the polyomials appearig i the square brackets of the above epressio. These are the Legere olyomials usually epresse i terms of the variable cosθ. They rea- 3 3 3 5 3 35 3 3 5 3 5 63 7 5 Sice θ i the potetial problem rages from π to π the rage of of iterest will be
-<<. Note that are eve fuctios while are o. Oe ca evelop a geeratig fuctio for these Legere polyomials startig with- Differetiatig oce with respect to we have- 3 ] [ ] [ or the equivalet form- [ ] ] 3 We thus have the geeratig formula for Legere polyomials- 3 [ This formula is easy to program startig with a. Oe fis, for eample, that- 5 5 35 3 6 6 6 A ifferetial equatio for ca be fou by eamiig the fuctio- ] [, F Differetiatig this fuctio oce yiels oce for,, 3, etc. yiels-, F
F, 6 a- F3, 3 3 From these results oe ca coclue that the ifferetial equatio goverig the Legere polyomials is- If we ow multiply this equatio by m a itegrate over the rage -<<, oe fis- m m Oe otes at oce that if iteger iffers from iteger m, that the left sie of this equality vaishes. Thus for [,m][,3] a [,] we have- 3 3 a [ 35 5 ] Whe m, however, the left sie oes ot vaish. For m,, 3, etc we fi m,,, etc 3 5 7 That is, the Legere polyomials are orthogoal i the rage -<<, with- m δ Here δ m is the Kroecker elta with value whe m a whe m. The fact that the s are orthogoal allows us to epa ay fuctio f i terms of them. Oe has- m
f C with C f For f, this epasio yiels the three term series- [7 35 ] Note by settig C /F[] we obtai the Legere trasform pair- F f with its iverse f [ ] F Ulike Laplace trasforms, Legere trasforms are geerally much more complicate. Oe of the simpler oes occurs for the Dirac elta fuctio. There F!, o / [ Γ.5 ], eve O ivertig, it prouces the ietity-! δ [ ][ ]! 5 7 6 65 3 6 595... 56 It takes about the first eight terms i this series to start seeig a fuctio approachig ifiity at a vaishig for all other i -<<. There are umerous itegrals ivolvig. Oe of the simplest is-!! 35..! as ca be establishe by lookig at the values for,,,3,. Aother itegral is-
which ca be establishe by otig the itegral has values.5 /3,.5 /5,.5 /7 for,,3, respectively. Aother iterestig itegral is- A Bπ where the values of A a B icrease i magitue with icreasig a have opposite sigs. We fi [A, B][3, -] for a for take o the values- A -79656579/969969 a B 3397/5 We thus fi the iterestig equality- π 33667 975367599 3397 where the first term alreay approimates π to some places a the seco term as a correctio of orer -. Sice the seco term i this epressio for π becomes progressively smaller with icreasig, it is clear that- π lim A B A 3 place accurate approimatio for π is fou at 5 a is give by the quotiet- A5 B 5 65367799756733659756 5939653937369379633595 We ca also geeralize the above itegral to fi- N k M arcta k Where N a M are umbers epeig o. Agai oe fis that a approimatio for arcta/k will be give by-
N arcta k M The covergece rate of this quotiet with icreasig is fou to be quite rapi. For oe alreay fis arcta/5 give accurately to places. A goo approimatios for l ca be gotte from the evaluatio - S 3 T l For large the term o the left ivie by T becomes small compare to the quotiet S /T. Thus at we fi the 3 igit accurate result- 676536763 l.69375599539735.. 5656533797 Fially let us look at a itegral represetatio for the Legere polyomials. Usig the seco orer ifferetial equatio for a a Euler Kerel -t as the startig poit, oe fissee our class otes i the above ODE course that- πi t t t This result is kow as the Schlaefli Itegral. By cotour itegratio arou the orer pole at, this itegral ca be evaluate by the Cauchy theorem to yiel! This represets the famous Roriques erivative formula for geeratig Legere polyomials. It is ot quite as coveiet for fiig values of at large via computer as is the geeratig formula give earlier. October 9