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Physics LibreTexts

1.14: Legendre Polynomials

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Consider the expression

(12rx+r2)1/2,

in which |x| and |r| are both less than or equal to one. Expressions similar to this occur quite often in theoretical physics - for example in calculating the gravitational or electrostatic potentials of bodies of arbitrary shape. See, for example, Chapter 5, Sections 5.11 and 5.12.

Expand the expression 1.14.1 by the binomial theorem as a series of powers of r. This is straightforward, though not particularly easy, and you might expect to spend several minutes in obtaining the coefficients of the first few powers of r. You will find that the coefficient of rl is a polynomial expression in x of degree l. Indeed the expansion takes the form

(12rx+r2)1/2=P0(x)+P1(x)r+P2(x)r2+P3(x)r3...

The coefficients of the successive power of r are the Legendre polynomials; the coefficient of rl, which is Pl(x), is the Legendre polynomial of order l, and it is a polynomial in x including terms as high as xl. We introduce these polynomials in this section because we shall need them in Section 1.15 on the derivation of Gaussian Quadrature.

If you have conscientiously tried to expand expression 1.14.1, you will have found that

P0(x)=1,P1(x)=x,P2(x)=12(3x21),

though you probably gave up with exhaustion after that and didn’t take it any further. If you look carefully at how you derived the first few polynomials, you may have discovered for yourself that you can obtain the next polynomial as a function of two earlier polynomials. You may even have discovered for yourself the following recursion relation:

Pl+1=(2l+1)xPllPl1l+1.

This enables us very rapidly to obtain higher order Legendre polynomials, whether numerically or in algebraic form. For example, put l=1 and show that Equation 1.12.4 results in P2=12(3x21). You will then want to calculate P3, and then P4, and more and more and more. Another way to generate them is form the Equation

Pl+1=12ll!dldxl(x21)l.

Here are the first eleven Legendre polynomials:

P0=1P1=xP2=12(3x21)P3=12(5x33x)P4=18(35x430x2+3)P5=116(63x570x3+15x)P6=116(231x6315x4+105x25)P7=116(429x7693x5+315x335x)P8=1128(6435x812012x6+6930x41260x2+35P9=1128(12155x925740x7+18018x54620x3+315x)P10=1256(46189x10109395x8+90090x630030x4+3465x263)

The polynomials with argument cosθ are given in Section 5.11 of Chapter 5.

In what follows in the next section, we shall also want to know the roots of the Equations Pl=0 for l>1. Inspection of the forms of these polynomials will quickly show that all odd polynomials have a root of zero, and all nonzero roots occur in positive/negative pairs. Having read Sections 1.4 and 1.5, we shall have no difficulty in finding the roots of these Equations. The roots xl,i are in the following table, which also lists certain coefficients cl,i, that will be explained in Section 1.15.

Roots of Pl=0

lxl,icl,i2±0.577 350 269 1901.000 000 000 003±0.774 596 669 2410.555 555 555 560.000 000 000 0000.888 888 888 894±0.861 136 311 5940.347 854 845 14±0.339 981 043 5850.652 145 154 865±0.906 179 845 9390.236 926 885 06±0.538 469 310 1060.478 628 670 500.000 000 000 0000.568 888 888 896±0.932 469 514 2030.171 324 492 38±0.661 209 386 4660.360 761 573 05±0.238 619 186 0830.467 913 934 577±0.949 107 912 3430.129 484 966 17±0.741 531 185 5990.279 705 391 49±0.405 845 151 3770.381 830 050 500.000 000 000 0000.417 959 183 688±0.960 289 856 4980.101 228 536 29±0.796 666 477 4140.222 381 034 45±0.525 532 409 9160.313 706 645 88±0.183 434 642 4960.362 683 783 389±0.968 160 239 5080.081 274 388 36±0.836 031 107 3270.180 648 160 69±0.613 371 432 7010.260 610 696 40±0.324 253 423 4040.312 347 077 040.000 000 000 0000.330 239 355 0010±0.973 906 528 5170.066 671 343 99±0.865 063 366 6890.149 451 349 64±0.679 409 568 2990.219 086 362 24±0.433 395 394 1290.269 266 719 47±0.148 874 338 9820.295 524 224 6611±0.978 228 658 1460.055 668 567 12±0.887 062 599 7680.125 580 369 46±0.730 152 005 5740.186 290 210 93±0.519 096 129 2070.233 193 764 59±0.269 543 155 9520.262 804 544 510.000 000 000 0000.272 925 086 7812±0.981 560 634 2470.047 175 336 39±0.904 117 256 3700.106 939 325 99±0.769 902 674 1940.160 078 328 54±0.587 317 954 2870.203 167 426 72±0.367 831 498 9980.233 492 536 54±0.125 233 408 5110.249 147 045 8113±0.984 183 054 7190.040 484 004 77±0.917 598 399 2230.092 121 499 84±0.801 578 090 7330.138 873 510 22±0.642 349 339 4400.178 145 980 76±0.448 492 751 0360.207 816 047 54±0.230 458 315 9550.226 283 180 260.000 000 000 0000.232 551 553 2314±0.986 283 808 6970.035 119 460 33±0.928 434 883 6640.080 158 087 16±0.827 201 315 0700.121 518 570 69±0.687 292 904 8120.157 203 167 16±0.515 248 636 3580.185 538 397 48±0.319 112 368 9280.205 198 463 72±0.108 054 948 7070.215 263 853 4615±0.987 992 518 0200.030 753 242 00±0.937 273 392 4010.070 366 047 49±0.848 206 583 4100.107 159 220 47±0.724 417 731 3600.139 570 677 93±0.570 972 172 6090.166 269 205 82±0.394 151 347 0780.186 161 000 02±0.201 194 093 9970.198 431 485 330.000 000 000 0000.202 578 241 9216±0.989 400 934 9920.027 152 459 41±0.944 575 023 0730.062 253 523 94±0.865 631 202 3880.095 158 511 68±0.755 404 408 3550.124 628 971 26±0.617 876 244 4030.149 595 988 82±0.458 016 777 6570.169 156 519 39±0.281 603 550 7790.182 603 415 04±0.095 012 509 8380.189 450 610 4617±0.990 575 475 3150.024 148 302 87±0.950 675 521 7690.055 459 529 38±0.880 239 153 7270.085 036 148 32±0.781 514 003 8970.111 883 847 19±0.657 671 159 2170.135 136 368 47±0.512 690 537 0860.154 045 761 08±0.351 231 763 4540.168 004 102 16±0.178 484 181 4960.176 562 705 370.000 000 000 0000.179 446 470 35

For interest, I draw graphs of the Legendre polynomials in figures I.7 and I.8.

Figure I.7. Legendre polynomials for even I
alt

Figure I.8. Legendre polynomials for odd I
alt

For further interest, it should be easy to verify, by substitution, that the Legendre polynomials are solutions of the differential Equation

(1x2)y2xy+l(l+1)y=0.

The Legendre polynomials are solutions of this and related Equations that appear in the study of the vibrations of a solid sphere (spherical harmonics) and in the solution of the Schrödinger Equation for hydrogen-like atoms, and they play a large role in quantum mechanics.


This page titled 1.14: Legendre Polynomials is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Jeremy Tatum via source content that was edited to the style and standards of the LibreTexts platform.

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