Exceptional Branch Points Without the Condition k>l

  • Anthony Tromba
Part of the Springer Monographs in Mathematics book series (SMM)


Now let Open image in new window be a nonplanar minimal surface in normal form at the exceptional branch point w=0 with \(k \not= l\) which does not satisfy k>l. We have
$$\hat{X}_w (w) = (A_1 w^n + A_2w^{n+1} + \cdots, R_m w^m + \cdots)$$
with m>n, \(A_{1} \not= 0\), \(R_{m} \not= 0\). As in Chap. 5 we introduce C j and \(C'_{j} \in \mathbb{R}^{2}\) by Set
$$\varTheta_j := (j-1) + (n+1);$$
$$w \hat{X}_w (w) = \biggl(A_1 w^{n+1} + \sum_{j> 1} C_j w^{\varTheta_j} + \sum _{j> 1} C'_j w^{\varTheta_j}, R_m w^{m+1} + \cdots\biggr).$$


Normal Form Holomorphic Function Minimal Surface Differential Geometry Periodic Function 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California at Santa CruzSanta CruzUSA

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