Proof of Liouville’s Theorem
Let’s Proof Liouville’s Theorem:
Consider f an entire function in a plane C.
Suppose a and b are arbitrary points.
Let f’= 0. Then f is a constant function.
To prove that f is a constant function, we need to show that f(a) = f(b) for all a, b ∈ C. Since C is a plane, which is path connected. Then we can choose a curve F: I➝C so that F(0) = a and F(1) = b.
Now,
∫xf'(x) dx = f(b)- f(a)
Since f’ = 0, f(b) = f(a)
Now,
Suppose |f(x)| ≤ P for all x ∈ C.
To prove ƒ is a constant, we only need to prove f'(z) = 0. Let a ∈ C.
Now, by using Cauchy-Integral Formula, rest of arithmetic operations can be performed.
In this way, you can prove the Liouville’s Theorem.
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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