d2j2003 said:I believe f(z) is analytic at infinity if f(1/z) is analytic at z=0
d2j2003 said:so we can say that since p(z)/z^n is analytic at infinity then it is analytic on the domain |z|>1 so according to the maximum modulus principle, p(z)/z^n can have no local maximum on domain (|z|>1) right?
d2j2003 said:and M is just a real number? it is not specified anywhere... and what about |z|=1 since it says on |z|≥1 and we have looked at |z|>1
The Maximum Modulus Principle is a theorem in complex analysis that states that the maximum value of a holomorphic function on a closed and bounded set in the complex plane occurs either at a boundary point or at a point inside the set where the function is not analytic.
The Maximum Modulus Principle is used to find the maximum value of a holomorphic function on a given set, by analyzing the behavior of the function at its boundary points. It is also used to prove the uniqueness of analytic functions, as well as to solve various problems in physics and engineering.
The Maximum Modulus Principle only applies to holomorphic functions, which are functions that are complex differentiable at every point within a given set. It also only applies to closed and bounded sets in the complex plane, and cannot be used for unbounded sets or functions that are not analytic.
The Maximum Modulus Principle has various applications in physics and engineering, particularly in the study of fluid dynamics, heat transfer, and electromagnetism. It is also used in the design and analysis of electronic circuits, as well as in the study of fluid flow in porous media.
Yes, there are several alternative principles in complex analysis that are similar to the Maximum Modulus Principle, such as the Minimum Modulus Principle, the Argument Principle, and the Cauchy Integral Theorem. These principles have different applications and may be used in different scenarios depending on the problem at hand.