Hill's equation by Magnus W., Winkler S.

By Magnus W., Winkler S.

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10) 16. Series of Lc~endre Polynomials. In certain problems of polcnlinl theory it is desirable to be able to express n given function in the form of 1\ series of Legendre polynomials. We Cfin readily show 1hn1 this is possible in the case in which the given function is n simple poly< nomial. u)+ wriltCIl C;I + es) 1'0(,1) It is ob\'ious that wc could proceed in this WlIr for [I polynominl of lilly gh'CIl degree" though if II were large the nrithmctic irwolvcd might becomc cumbersome.

1. p, y: '] :I/'("ll'(1I1 1 6. 6In~+idr(ifJ+i6lrUfJ+ltl CIIAPTER III LEGENDRE FUNCTIONS 13. Lcgcndre Polyllomhlls. nliollal potcntial at the point P due to n unit mass situatcd at the point A. and that this must be n particular solution of Laplacc's equation. In some eireumstnnecs it is desirable to expand 'I' in y' :2)1 is the dislnnee powers of r or r l where r = (r + + p , 11 01 0 " Fig. 4 A of I~ from O. the onglll of coonlinutes. This expansion can be obluint:d by the usc of Tnylor's theorem for functions of thrce \·nrillbles but it is much marc suitnble to '" 113 " LEGENDRE FUNCTIONS introduce the angle 0 between the directions OA.

X ill O,> (J. 1). :r, -, - - , - - , - a: I-a: oX x-l «+ the equation trnnsforms to one of Lhc Sfime type (but, of course, with differcnt parnlllclcrs). 0) - two for eneh independent vllrinble - each eOll\'ergent within the unit circle. Any olle of these enn be expressed in tenns of ty,·o fundamental solutions. In addition twelvc more solutions of the kinds F 1(Y-:l.

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