convergence for Bateman forms
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Short note dated 3.26.05 by Phil (PhL) listing the convergence conditions for the Bateman forms of the Legendre functions P and Q, keyed to equation numbers for each. For each argument such as (1-z)/2, 1-z^2 or z^2 it states whether convergence is inside, outside, left or right of a plotted boundary, with test points. It mentions Maple implicitplot plots and a correction: for the ratio cases the boundary reduces to the segment -1 to 1. Some symbols are garbled in the extraction.
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Convergence for Bateman forms.doc PhL 3.26.05
Condition for convergence is what is stated to the right. So "inside" means the form converges inside the figure drawn. The left equation number is for P, the right for Q.
I made these plots like this:
z := x+I*y;
implicitplot(abs( (1-z)/2 ) = 1, x=-4..4,y=-4..4 , scaling = CONSTRAINED, grid = [100,100]);
(1-z)/2 (14) (32) inside
2/(1-z) (19) (37) outside
(1+z)/2 (15) (33) inside
2/(1+z) (18) (36) outside
(z-1)/(z+1) (16) (34) right try z = 1
(z+1)/(z-1) (17) (35) left
Note added: |z-1| = |z+1| defines the imaginary axis so the above is correct,
(z-/(z+ (27) (45) try z = 2
(z+/(z- (31) (49)
Note added: Plot for the above cases was done wrong. Let z = chξ to find that ratio = e-2ξ. Then you need that |e-2ξ| = 1 which means ξ = iθ which means z = ch(ξ) = cosθ which lies on -1 to 1. I gave up on what the "sides are" in this case. If you want to pursue that topic, see "convergence of Bateman forms. mws" .
1-z2 (20) (38) inside
1/(1-z2) (21) (39) outside
z2 (22) (40) inside
1/z2 (23) (41) outside
1-1/z2 (24) (42) left/right try z = ±2
z2/(z2-1) (25) (43) middle try z = 0
The real intersection is 1/ = .707.
[(2/(z+] (28) (46) right and z=-1 try z = 0 =>div
[(z+/(2] (30) (48) left
(-z+/[2] (26) (44) right try z = 0
(2/(-z+] (29) (47) left and z = 1 try z=0