Frequently Asked Questions on Liouville’s Theorem
What is Liouville’s Theorem in Complex Analysis?
Liouville’s theorem in complex analysis states that, “a bounded entire function is a constant function.”
What are Requirements of Liouville’s Theorem?
A function that satisfies Liouville’s Theorem must follow the condition that the function has to be an entire function as well as bounded for all values of z in z-plane.
Can Liouville’s Theorem only be applied to complex numbers?
Yes, Liouville’s Theorem is particularly based on complex analysis and entire functions.
Is Liouville’s Theorem Used for Real-World Applications?
Yes, its applications expand to various fields like physics, engineering, economics etc.
Liouville’s Theorem is named after which Mathematician?
Joseph Liouville a famous French mathematician is the one after which Liouville’s theorem is named.
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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