By Robert P. Gilbert
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Extra resources for Function Theoretic Methods in Partial Differential Equations
Sample text
And g(z) has singularities at pl, j2, . . , then the singularities of F(z) = Canb, z" are to be found at the points u, p, . To prove this theorem we first consider the representation for F(z) in a sufficiently small neighborhood of the origin, where 9 is a simple contour lying in the annulus IzI R'-' < l[l < R. This representation may be continued analytically along a curve y to zl, provided no point of y coincides with a singularity of the integrand on the path of 3. SINGULAR POINTS OF HOLOMORPHIC FUNCTIONS 23 integration.
The constant E may be further chosen so that arg[s] = arg[a,]. Further, let q’ be taken to lie in the range, p”(ao)- lzl*l < q’ < p”(ao- E ) - lzl*l and let B be the bicylinder These sets are indicated in Fig. 3. 2] it is w,ta ( p real) aot6 a0 ag-e I ( X real) FIGURE 3 clear that if for a = a, zl0 is not a singular point of Fa(zl),then Fa(zl) has a continuation into a disk = {zl Izl - zlol < q ‘ } where q‘ is chosen sufficiently small. 18) then it is clear that the function F,(zl) can be holomorphjcally continued into the bicylinder 8, providing b c Y u YJl.
We turn our attention next to the function R ( z ; w), which we define as R ( z ; w ) = F ( z ; w) n [w - -1 Wk(Z)l) (k11 . For each fixed (z') E A("), R(z("; w ) is holomorphic in w for IwI I a with ) = 1,2, . , m). These points must lie the exception of the m-points, W , ( Z ( ~ )(s strictly inside the disk { IwI < a } , since for IwI = a we have chosen A(") such that F(w; z ) # 0. We may conclude from this that the only singularities of Q(z; w ) are removable, and that at these points the function R(z; w ) may be defined by its Cauchy integral, .