proof of estimating theorem of contour integral
WLOG consider a parameterization of the curve along which the integral is evaluated with . This amounts to a canonical parameterization and is always possible. Since the integral is independent of re-parameterization11apart from a possible sign change due to exchange of orientation of the path the result will be completely general.
With this in mind, the contour integral can be explicitly written as
where is the arc length of the curve .
The axioms are easy to verify:
With all this in mind, equation 1 can be written as
Where by definition is the norm associated with the inner product defined previously.
Using Cauchy-Schwarz inequality we can write that
But since by assumption the parameterization is canonic, .
On the other hand , where for every point on .
The previous paragraphs imply that
which is the result we aimed to prove.
Cauchy-Schwarz inequality says more, it also says that where is a constant.
So if then , where is a constant. If is a canonical parameterization and we get the absolute modulus (which must be constant) and all that remains is to find the phase of which must also be constant.
|Title||proof of estimating theorem of contour integral|
|Date of creation||2013-03-22 15:46:02|
|Last modified on||2013-03-22 15:46:02|
|Last modified by||cvalente (11260)|