Liouville’s Theorem and Fundamental Theorem of Algebra
The differences between Liouville’s Theorem and Fundamental Theorem of Algebra is added in the table below,
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Difference Between Liouville’s Theorem and Fundamental Theorem of Algebra |
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Liouville’s Theorem |
Fundamental Theorem of Algebra |
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Focuses on an entire function in complex analysis. |
Focuses on various characteristics of polynomial functions. |
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Entire bounded functions must be constant. |
Roots of non-constant polynomials must be in the complex plane. |
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Refers to functions defined on the entire complex plane |
Specifically, it belongs to the polynomial equations in 1 variable |
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Restricts the behavior of entire functions |
Define the existence of roots for non-constant polynomials |
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Calls out the behavior related to boundedness |
Determines the roots and factorization of polynomials |
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Liouville’s Theorem
Liouville’s Theorem implies that every entire bounded function must be constant. In other words, according to Liouville’s theorem, “An entire (that is, holomorphic in the whole complex plane C) function cannot be bounded if it is not constant.” This theorem of complex analysis was given by a French mathematician, Joseph Liouville.
In this article, we discuss what Liouville’s theorem is, its application, and its importance in complex analysis.
Table of Content
- What is Liouville’s Theorem?
- Proof of Liouville’s Theorem
- Corollaries of Liouville’s Theorem
- Applications of Liouville’s Theorem
- Liouville’s Theorem and Fundamental Theorem of Algebra
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