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 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: 334 LEGENDRE 8.5.2 (22-1) ==(V+p) (v-p+1)(22--l)~P\$-l(z) dz --ELzp; (4 Varying Degree 8.5.3 (v- p + l)P\$ + 1 ( 2 ) = (2V + 1) zp: ( 2 ) - (. + lU>P\$ - 1 ( 4 8.5.4 (2-1) ' m = v z P \$ ( z ) - (~+p)P\$-~(z) dz Varying Order and Degree 8.5.5 P\$ + 1 ( 2 ) =P\$ - 1 ( 2 ) + (2 v + 1) (2' - 1) +P\$ - ( 2 ) 8.6. Special Values X=O 8.6.1 p: (0) =2r,-f cos [3T(V+p)J r (+v+ ip++)/r (3.- +p+i) 8.6.2 &:(O)= -2~-%4 sin [+T(v+~)] r (+v+ +p+ \$)/r ( i v - i p + l ) 8.6.3 r@?pl=o= 2r+%-f sin [+,(v+p)] r (+++p+ i ) / r (++p++) 8.6.4 8.6.6 8.6.7 8.6.8 p=m=l, 2 , 3 ' . . . d"P,(z) fi( 2 ) = (22- 1) hrn ~ dz" ' dmPy(x) e (2) = (- 1 ) " ( 1 - x2) trn ___ dx" + [ 2 + (22- 1) 1/21 - "-f - 'UNCTIONS 8.6.9 1/2 (zZ-1)-1/4 P,"z)= - { [ z + (22- 1) 1/21 Y+ f (3 2V+l -[z+(z"l ) I-"-fl 8.6.10 &!( z ) = i ( 3,) 1/2( 22- 1) --1/4[ 2 + (22- 1) 1/21 - v - 4 8.6.11 * 8.6.12 Pi(cos e ) = ( + T ) - f (sin e)-* cos [(V++)e] 8.6.13 &t(cos e) = - (+T)+ (sin e) -1 siii [ (V+:a)e] 8.6.14 P;~(COS e)=(3T)-+(V++)-1(sin e)-4 sin [(v++)e] 8.6.15 * &;f(cos e ) = ( 2 ~ ) *(2~+1)-l(sin e) -f cos [ (v+i)e] p=-V 8.6.16 p=O, v=n (Rodrigues' Formula) 1 d"(Z2-l)" Pfl(Z) =- 2"n! dz" 8.6.18 w- 1 (x) = 0 *See page XI. 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.