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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 | MEDIUM | LARGE Below is the OCR-scanned text from this page: 589 17. Elliptic Integrals Mathematical Properties 17.1. Definition of Elliptic Integrals If R(z, y) is a rational function of x and y, where y2 is equal to a cubic or quartic polynomial in x, the integral 17.1.1 \$m,Y)dX is called an elliptic integral. The elliptic integral just defined can not, in general, be expressed in terms of elementary functions. Exceptions to this are (i) when R(z, y) contains no odd powers of y. (ii) when the polynomial y2 has a repeated factor. We therefore exclude these cases. By substituting for y2 and denoting by p,(x) 8 polynomial in x we get where Rl(s) and R2(x) are rational functions of 5. Hence, by expressing &(x) as the s u m of 8 poly- nomial and partial fractions + &B8s [ (Z-C) 'yl-'dz Reduction Formulae Let 17.1.2 y2= a d + a d +a&+ w+a4 (la01 + la11 4 ( l ~ O I + l ~ l l +O) =~,(z-c)~+ ~!(z-c)'+ b z ( ~ - ~ ) ' + ba(3-c) + b 4 17.1.3 18= ~'~-'d;c, J8= [ ~ ( x - c ) ~ ] - ' ~ x S S By integrating the derivatives of yx8 and ~ ( Z - C ) - ~ we get the reduction formulae 17.1.4 (8 + 2)aoL+3+ 3 a1 (28 + W a + 2 + ads + 1 )Il+' + 3 q ( 2 s + l ) l , + ~ a ~ l , _ ~ = s ~ y (s=O, 1 , 2, . . .) 'See [17.71 22.72. 17.1.5 (2-8)bJ8-3+ # b1 (3-28)J8-2+ b2( 1 -8)JS-I +* b3( 1-28>J1-8b4~8+1=y(x-c)-~ (s=1, 2, 3, . . .) By means of these reduction formulae and cer- tain transformations (see Examples 1 and 2) every elliptic integral can be brought to depend on the integral of a rational function and on three canonical firms for elliptic integrals. 17.2. Canonical Forms Definitions 17.2.1 m=sinz a; m is the parameter, 17.2.2 x=sin p=sn u 17.2.3 COB p=cn u 17.2.4 a is the m id r angl (1-m sina p)*=dn u=A(p), the delta amplitude 17.2.5 p=arcsin (sn u)=am u, the amplitude Elliptic Integral of the First Kind 17.2.6 ~(p\a)=F(c(plm)=S'(l-sinPa 0 sin20)-W =I[ (1 - t (1 - mt ]-+dt 17.2.7 dw=u =s,' Elliptic Integral of the Second Kind 17.2.8 E (p\a) = E (u 1 m) = ( 1 - t Z, - * ( 1 - rn t ') *dt =s,'(1 -sin2 a sin2 e)*d 17.2.9 17.2.10 = p n ' w dw 17.2.11 =mlu+m~ucnz w dw 0 I 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.