Recent content by AcC

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    Rouche's Theorem: Find Zeros of f(z)=z^9-2z^6+z^2-8z-2 Inside Unit Circle

    Homework Statement Find the number of zeros of the folowing polynomial lying inside the unit circle, f(z)= z^9 - 2z^6 + z^2 - 8z - 2 The Attempt at a Solution Rouche's Theorem says if f and g differentiable which contains a simple loop s and all points inside s...
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    Is f constant if limf(z) exists and is nonzero as z tends to z0?

    Take limit e^z as z→-infinity it equal to zero, so it not satisfy the statement. I proved this statement with using Liouville Theorem.
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    Complex Analysis: Showing f is a Polynomial of Degree n

    Thanks a lot for hint..:) I have solved this as below; since f is analytic on C so f is differentiable on C, if we use Cauchy's Estimate then we find |f^{(k)}(z)|<=M.|z|^n.k!/|z|^k for k>=n if take lim as z→0 then we find...
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    Complex Analysis: Showing f is a Polynomial of Degree n

    Homework Statement Let f be analytic through C, suppose that |f(z)|<=M|z|^n for a real constant M and positive integer n. Show that f is a polynomial function of degree less than n.
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    Is f constant if limf(z) exists and is nonzero as z tends to z0?

    Yes you are true!?? but I found this question in complex analysis book, I tried to solve but I hadnt thought that it could be false..!
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    Is f constant if limf(z) exists and is nonzero as z tends to z0?

    Homework Statement Let f:C\rightarrowC be differentiable, with f(z)\neq0 for all z in C. Suppose limf(z) is exist and nonzero as z tends to z0. Prove that f is constant.
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    Proving Connectivity of B & cl(A)

    Thanks grey_earl but I want to show if A is connected set and A ⊂ B ⊂ cl(A) then B is connected.. I have to show that ""Suppose B=G∪H is a disconnection of B, then using the fact that B ⊂ cl(A) prove that contradict the given A is connected"" How can I show this?
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    Proving Connectivity of B & cl(A)

    Homework Statement Let A be connected subset of X and let A ⊂ B ⊂ cl(A). Show that B is connected and hence, in particuar, cl(A) is connected. Hint: (Use) Let G∪H be a disconnection of A and let B be a connected subset of A then we see that either B∩H=∅ or B∩G=∅, and so either B⊂G or B⊂H...