Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (AMS55)
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8 . Legendre Functions IRENE A . STEGUN Con tents Page Mathematical Properties . . . . . . . . . . . . . . . . . . . . 332 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 332 8.1. Differential Equation . . . . . . . . . . . . . . . . . 332 8.2. Relations Between Legendre Functions . . . . . . . . . . 333 8.3. Values on the Cut . . . . . . . . . . . . . . . . . . . 333 8.4. Explicit Expressions . . . . . . . . . . . . . . . . . . 333 8.5. Recurrence Relations . . . . . . . . . . . . . . . . . 333 8.6. Special Values . . . . . . . . . . . . . . . . . . . . 334 8.7. Trigonometric Expansions . . . . . . . . . . . . . . . 335 8.8. Integral Representations . . . . . . . . . . . . . . . . 335 8.9. Summation Formulas . . . . . . . . . . . . . . . . . 335 8.10. Asymptotic Expansions . . . . . . . . . . . . . . . . 335 8.11. Toroidal Functions . . . . . . . . . . . . . . . . . . 336 8.12. Conical Functions . . . . . . . . . . . . . . . . . . . 337 8.13. Relation to Elliptic Integrals . . . . . . . . . . . . . . 337 8.14. Integrals . . . . . . . . . . . . . . . . . . . . . . . 337 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . 339 8.15. Use and Extension of the Tables . . . . . . . . . . . . 339 References . . . . . . . . . . . . . . . . . . . . . . . . . . 340 Table 8.1. Legendre Function-First Kind Pn(x) (251) . . . . . . . 342 ~=0(.01)1, n=0(1)3. 9. 10. 5-8D Table 8.2. Derivative of the Legendre Function-First Kind Pk(z) ($51) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 Table 8.3. Legendre Function-Second Kind Qn(z) ($21) . . . . . . 346 ( ~ 5 1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 Table 8.5. Legendre Function-First Kind P, (z) ($21) . . . . . . . 350 ($21) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 Table 8.7. Legendre Function-Second Kind Qn(z) (~21) . . . . . . 352 (521) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 ~=0(.01)1, n=1(1)4. 9. 10. 5-7D $=0(.01)1, n=0(1)3. 9. 10. 8D Table 8.4. Derivative of the Legendre Function-Second Kind Qk(z) ~=0(.01)1, n=0(1)3, 9. 10. 6-8D 2=1(.2)10, n=0(1)5. 9. 10. exact or 6s Table 8.6. Derivative of the Legendre Function-First Kind P:(z) 2=1(.2)10, n=1(1)5. 9. 10, 6 s 2=1(.2)10, n=0(1)3, 9. 10. 6 s Table 8.8. Derivative of the Legendre Function-Second Kind Q:(z) 2=1(.2)10, n=0(1)3. 9. 10. 6s The author acknowledges the assistance of Ruth E . Capuano. Elizabeth F . Godefroy. David S . Liepman. and Bertha I1 . Walter in the preparation and checking of the tables and examples . 1 National Bureau of Standards . 331