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Understanding the Maximum Modulus Principle - Testbook.com

Last Updated on Jul 31, 2023
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Delving into the world of complex analytic functions, we come across the maximum modulus principle . This principle, or theorem, posits that the highest value of the modulus of a function defined on a bounded domain can only occur on the boundary of that domain. If the function's modulus reaches its peak value within the domain, the function must be constant.

In essence, the maximum modulus principle describes the local maximum of an analytic function within a domain. The highest value can only be achieved on the boundary unless the function is constant.

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Explaining the Maximum Modulus Principle

Consider G ⊂ C (where C denotes the set of complex numbers) as a bounded and connected open set. If f is an analytic function defined on G, the maximum value of |f(z)| occurs on G, not inside G, unless f is a constant function.

This means if M is the maximum value of |f(z)| on and within G, then unless f is constant, |f(z)| < M for every point z within G.

Thus, the maximum modulus principle concludes that a non-constant analytic function's absolute value on a connected open set G ⊂ C cannot have a local maximum point in G.

Demonstrating the Maximum Modulus Principle

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Practical Examples of the Maximum Modulus Principle

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Frequently Asked Questions

The maximum modulus principle for complex analytic function states that the absolute value of a non-constant analytic function on a connected open set G ⊂ C cannot have a local maximum point in G.

The maximum modulus principle requires that the given function be analytic within the given domain.

The maximum modulus principle is used to prove many important theorems in complex analysis: the fundamental theorem of algebra, Schwarz’s Lemma, Borel-Caratheodory theorem, Hadamard’s three-line theorem.

According to the minimum modulus principle, the absolute minimum value of an analytic function defined on a bounded domain occurs on the boundary of the domain. The minimum modulus principle can be proved using the maximum modulus principle for 1/f.
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