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17. Elliptic Integrals
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
is called an elliptic integral.
The elliptic integral just defined can not, in
general, be expressed in terms of elementary
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
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
(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.
(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
m=sinz a; m is the parameter,
17.2.2 x=sin p=sn u
17.2.3 COB p=cn u
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
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
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