The Maximum Modulus Principle in complex analysis states that if a function is analytic (holomorphic) and non-constant within a domain, the maximum value of its modulus (absolute value) will always occur on the boundary of that domain. If a maximum modulus is found inside the domain, the function must be constant within that domain. [1, 2]
In simpler terms:
Imagine a function defined on a region. The maximum modulus principle says that if you're looking for the largest value of the "size" of the function (its absolute value), you only need to check the edges (boundary) of the region. If you find the largest value inside the region, it means the function isn't changing (it's constant). [1]
Key points:
- Analytic/Holomorphic: The function must be analytic, meaning it has a derivative at every point within the domain. [1, 2, 3, 4]
- Non-constant: The function cannot be a constant function (e.g., f(z) = 5 for all z). [1, 2]
- Domain: The principle applies to a region in the complex plane. [1]
- Boundary: The boundary of a domain is the edge or the outer limit of the region. [1, 5, 6]
Mathematical Formulation:
If f(z) is analytic and non-constant in a domain D, then the modulus |f(z)| cannot attain its maximum value at any interior point of D. If |f(z)| does attain its maximum value at a point z₀ in D, then f(z) is constant in D. [1, 2, 7]
This video explains the Maximum Modulus Principle and provides examples: https://www.youtube.com/watch?v=BUrMrUF_IVU&pp=0gcJCfwAo7VqN5tD
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