proof of maximal modulus principle
Suppose the maximum of is attained at in the interior of .
By definition there will exist such that the set .
A canonical parameterization of is , for .
Since is a maximum, the last inequality must be verified by having the equality in the verified.
The proof of the estimating theorem of contour integral (http://planetmath.org/ProofOfEstimatingTheoremOfContourIntegral) implies that equality is only verified when
So is holomorphic and constant on a circumference. It’s a well known result that if 2 holomorphic functions are equal on a curve, then they are equal on their entire domain, so is constant.
to see this in this particular circumstance is using equation 1 to calculate the value of on a point interior different than . Bearing in mind that is constant in the formula reads . So is really constant in the interior of and the only holomorphic function defined in that is constant in the interior of is the constant function on all .
Thus if the maximum of is attained in the interior of , then is constant. If isn’t constant, the maximum must be attained somewhere in , but not in its interior. Since is compact, by definition it must be attained at .
|Title||proof of maximal modulus principle|
|Date of creation||2013-03-22 15:46:15|
|Last modified on||2013-03-22 15:46:15|
|Last modified by||cvalente (11260)|