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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: LEGENDRE FUNCTIONS 333 Wronskian 8.1.8 8.1.9 W{Pn(z>, Qn(z)} =--(~'-l)-' 8.2. Relations Between Legendre Functions Negative Degree 8.2.1 PLty--l(z) =P:(z) 8.2.2 ~ f ~ - ~ ( 2 ) = f - r e f f i r cos v7rEtz) - +&:(z) sin [7r(v+~Co11/sin [&-dl Negative Argument (\$zZO) n 8.2.3 L P: (- z ) = eFf '*P: ( z ) -; e -'P sin [ ~ ( v + p) 1 Q: ( z ) 8.2.4 Q:(-z)=-e**v'" Q Y ( z 1 Negative Order 8.2.5 8.2.6 Degree p+ 3 and Order v + 3 * %%>o (Upper and lower signs according as YzsO.) * [ - 61:(z+W 8.3.3 =ir-le,-i@~ e t l P -effp*&:(s-iO)] * 8.3.4 @! (2) = +e -+*[ e-tipr&c (z + iO) + e i f p * @ (s - i O ) ] (Formulas for P:(z) and @(z) are obtained with the replacement of 2-1 by (l-s)e*la, ( 9 - 1 ) by (1--z2)e*i*, z + l by z+1 for z=zrfiiO.) 8.4. Explicit Expressions (Z=COS e) 8.4.1 Po(z) = 1 Po(z) = 1 8.4.2 =sF(+, 1; Q ; 22) 32 2 -- 8.5. Recurrence Relations (Both P: and &: satisfy the same recurrence relations.) 8.5.1 Varying Order P;+1(2)=(2-1)-*{ (Y-p)zP:(z)-((v+p)P:-l(z) 1 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.