Complex analysis modulus theorem
How do you prove the maximum modulus theorem?
Proof of Maximum Modulus Principle
The function f attains maximum value M at a point within or on G; we claim that this point only lies on G and not inside G.
Suppose, if possible, this maximum value M of f occurs at any point z = a inside G.
Then, maxf(z) = f(a) = M and f(z) ≤ M for all z within G..
What is the minimum modulus theorem?
Theorem (Minimum Modulus Theorem).
If f is holomorphic and non- constant on a bounded domain D, then f attains its minimum either at a zero of f or on the boundary.
Proof.
If f has a zero in D, f attains its minimum there..
What is the modulus of a complex function?
The modulus of a complex number is the square root of the sum of the squares of the real part and the imaginary part of the complex number.
It can be calculated using the formula z = √(x2 + y2)..
- Identity theorem Let f be holomorphic functions on some domain Du228.
- C, and let S be the set of all zeros of f which has a limit point in D.
Then f is identically zero in D.
Proof in short: Let a be limit point of S this means that there exist a sequence an of zeros such that an→a as n→∞.
- The maximum modulus principle has many uses in complex analysis, and may be used to prove the following: The fundamental theorem of algebra.
Schwarz's lemma, a result which in turn has many generalisations and applications in complex analysis. - Theorem (Minimum Modulus Theorem).
If f is holomorphic and non- constant on a bounded domain D, then f attains its minimum either at a zero of f or on the boundary.
Proof.
If f has a zero in D, f attains its minimum there.
According to the Maximum Modulus Theorem in complex analysis, if a non-constant function/(z) is continuous on a closed bounded region R and is analytic at every interior point of R, then the maximum value of \f (z) | in R must occur on the boundary of R.
In mathematics, the maximum modulus principle in complex analysis states that if f {\displaystyle f} f is a holomorphic function, then the modulus | f Formal statementMinimum modulus principleUsing Gauss's mean value
On when a family of real, continuous functions has a uniformly convergent subsequence
The Arzelà–Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence of a given family of real-valued continuous functions defined on a closed and bounded interval has a uniformly convergent subsequence.
The main condition is the equicontinuity of the family of functions.
The theorem is the basis of many proofs in mathematics, including that of the Peano existence theorem in the theory of ordinary differential equations, Montel's theorem in complex analysis, and the Peter–Weyl theorem in harmonic analysis and various results concerning compactness of integral operators.
Mathematical inequality
Bernstein's theorem is an inequality relating the maximum modulus of a complex polynomial function on the unit disk with the maximum modulus of its derivative on the unit disk.
It was proven by Sergei Bernstein while he was working on approximation theory.
Uniqueness theorem in complex analysis
In mathematics, in the area of complex analysis, Carlson's theorem is a uniqueness theorem which was discovered by Fritz David Carlson.
Informally, it states that two different analytic functions which do not grow very fast at infinity can not coincide at the integers.
The theorem may be obtained from the Phragmén–Lindelöf theorem, which is itself an extension of the maximum-modulus theorem.
