Complex Variable Homework 12(due in class Tuesday December 8)1. For jzj < 1 letu (z) = Im"1+z1 z2#:Show thatlim u reir!1=0for all 2 R. Does this violate the uniqueness of solution to Dirichletproblem? Why?2. Let (r; ) be the polar coordinate, show that11u = urr + ur + 2 u :rrIn particular, if u = f (r) is a harmonic function, show thatu = c1 log r + c2 :3. Let u be a C 2 function on the ball BR or annulus fz : R1 < jzj < R2 g, andZ1u (r; ) =uds:2 r @BrShow that u is C 2 on the same domain andZ1u (r; ) =uds =2 r @Bru (r; ) :In particular, if u is harmonic, then so is u.4. Prove the unique continuation property for harmonic functions, namely ifis open and connected, u is a harmonic function on such that u = 0on an open subset 1, then u is identically zero.5. Let u be a harmonic function on the unit ball B1 with kuk1 = supz2B1 ju (z)j <1, show thatjru (0)j c kuk1 :Here c is an absolute constant.16. Show that any bounded harmonic function on the whole plane must be aconstant function.7. Prove the removable singularity theorem for harmonic function. Namelyif u is a bounded harmonic function on the punctured disk B1 n f0g, thenu can be extended to a harmonic function on B1 .8. If f is holomorphic in fz 2 C : jzj > 1g such thatRe f (z)= 0;z!1zlimshow that limz!1 f (z) exists.2
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