Fundamental Theorem of Algebra in Complex Analysis

 In this page, we are going to analysis the important topic or theorem in Complex Analysis.  Fundamental Theorem of Algebra in Complex Analysis is a very important topic to understand that can enhance your Higher Mathematics knowledge.


Fundamental Theorem of Algebra in Complex Analysis




Statement: If f(z) =  a0zn + a1zn-1 + ... + an , a0 ≠ 0 be a polynomial in z, then f(z) = 0 has at least one root in z-plane.

Proof: If possible, let f(z) = 0 has no root in z-plane.
Therefore, f(z)  ≠ 0 for any value of z in z-plane.

We define,

because, f(z) is analytic and f(z)  ≠ 0
therefore, F(z) is analytic for all values of z.

Also, |F(z)| --> 0 as |z| -->  ∞
 therefore, F(z) is bounded
therefore, By Liouville's Theorem, F(z) is constant.

⇒ f(z) is constant, which is contradiction [because, f(z) is not constant polynomial]

therefore, f(z) = 0 has at least one root in z-plane.


Explanation in YouTube:


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