Suppose T is the family of all complex polynomials k=YJ f{z) = a,z' such that jafcl ^ 10̂ ' for fc =...

Question

Suppose T is the family of all complex polynomials
k=YJ
f{z) = a,z'
such that jafcl ^ 10̂ ' for fc = 0 , 1 , 1 7 . Show that J " is a normal family.
Let / be analytic on {2; : 0 < \z\ 2} . and suppose that for n = 0 ,1 , 2....
[ z^f{z)dz = 0.
Show that / has a removable singularity st 2; = 0. /
Let f is an analytic function on the open unit disk D. Suppose there exist^infinitety many
distmet z,ED,k^ 1,2,3,... such that f{zk) - 0. Does tlxis imply that j =
ove or
give a counterexample.

005_m6c8azo-nv040m2t.pdf

Robert · Accepted Answer

Yes, Problem is the convergence of the infinite product.
So lets have this product
f(z) =
1
z + 1
∞∏
n=1
(
z − (−1 + 1
n
)
)
For any z, when z = −1 + 1n , f(z) = 0.
Hence this function has infinite zero. This function is not analytic at z = −1.
As the term z + 1 is in the denominator.

Suppose T is the family of all complex polynomials k=YJ f{z) = a,z' such that jafcl ^ 10̂ ' for fc = 0 , 1 , 1 7 . Show that J " is a normal family. Let / be analytic on {2; : 0

Solution

Answer To This Question Is Available To Download

Related Questions & Answers

Submit New Assignment