By Lyn C. Thomas, David B. Edelman, Jonathan N. Crook
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We will use both expressions for dh. (x + x-l)u-k = t-1 - z + (x - x-l) = -z-‘(z - X)(Z + x-l), r/4 I o < 7r/2, minus the points (2,~) = (2, oo), (-x-‘,oo)(oo,O). at (oo,O) is the middle end. 6 The Logarithmic Growth Rates We proceed to determine the logarithmic growth rates of the catenoid ends. 5(vi), we know that this growth rate is equal to minus the residue of dh at the end. At the bottom end, where z = x, and at the top end, where z = -x-l, dz is regular, while at the middle end, where 2 = 00, dz/z has a simple pole with residue -1.
On complete minimal surfaces with finite Morse index in 3-manifolds. Invent. Math. 82, 121-132 (1985), Zbl. 53038 23. : The structure of complete stable minimal surfaces in S-manifolds of non-negative scalar curvature. Comm. Pure Appl. Math. 33, 199-211 (1980), Zbl. 53060 24. Cohn-Vossen. Arch. Ration. Mech. Anal. 61 (2), 141-152 (1976), Zbl. 53002 I. Complete Embedded Minimal Surfaces of Finite Total Curvature 91 25. : Index and total curvature of complete minimal surfaces. Proc. Symp. Pure Math.
Two vertices -e@ and -ie8, We will show that, in fact, p = CY, where a is the angle defined by the mapping u. To do this we will construct, from T and u, the same conformal map to the lower semi-disk. First of all (T)2 maps our quadrilateral to a semi-disk, which we rotate by -e2@ so that -e-@ -+ -1, -iewifl + +l, of course 0 -+ 0, and -1 --+ -e2@. In particular, -e22p(T)2 takes the r/kvertex to -e+2ip. Turning now to u, note that uk maps the quadrangle to the upper semi-disk. Composition of uk with the linear fractional transformation V(C) = &$& not only takes the upper semi-disk to the lower semi-disk, but on the three vertices opposite also has the property that v(u”) = -e2ifl(T)2 everywhere, in particular at the the r/k-vertex.
Credit Scoring & Its Applications by Lyn C. Thomas, David B. Edelman, Jonathan N. Crook