New Online Book! Handbook of Mathematical Functions (AMS55) Conversion & Calculation Home >> Reference Information Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (AMS55) Purchase the electronic edition of this book in Adobe PDF format! FIRST | PREVIOUS | NEXT | LAST | CONTENTS | PAGE | ABOUT | SMALL | MEDIUM | LARGE Below is the OCR-scanned text from this page: 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 The page scan image above, and the text in the text box above, are contributions of the National Institute of Standards and Technology that are not subject to copyright in the United States. ©2000 ConvertIt.com, Inc. All rights reserved. Terms of Use.