proof of estimating theorem of contour integral

WLOG consider g(t): a parameterization of the γ curve along which the integralPlanetmathPlanetmath is evaluated with |g(t)|=1. 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

γf(z)𝑑z=0Lf(g(t))g(t)𝑑t (1)

where L is the arc lengthMathworldPlanetmath of the curve γ.

Consider the set of all continuous functionsMathworldPlanetmath [0,L] as a vector spaceMathworldPlanetmath22axioms are trivial to verify, we can define an inner productMathworldPlanetmath in it via

f,g=0Lf(t)g¯(t)𝑑t (2)

The axioms are easy to verify:

  • k1a1+k2a2,a3=0L(k1a1(t)+k2a2(t))a3¯(t)𝑑t=k1a1,a3+k2a2,a3

  • a,b=0La(t)b¯(t)𝑑t=0Lb(t)a¯(t)¯𝑑t=0Lb(t)a¯(t)𝑑t¯=b,a¯

  • a,a=0La(t)a¯(t)𝑑t=0L|a(t)|2𝑑t0 since the integrand is a non-negative (real) function, and 0 iff |a|2=0 everywhere in the interval, that is: a,a=0a=0

With all this in mind, equation 1 can be written as

γf(z)𝑑z=fg,g¯ (3)

Where by definition f=<f,f> is the norm associated with the inner product defined previously.

Using Cauchy-Schwarz inequality we can write that

|fg,g¯|fgg¯ (4)

But since by assumption the parameterization g is canonic, g¯=g=0L1𝑑t=L.

On the other hand fg=0Lf(g(t))f¯(g(t))𝑑t0LM2𝑑t=ML, where |f(g(t))|M for every point on γ.

The previous paragraphs imply that

|γf(z)𝑑z|ML (5)

which is the result we aimed to prove.

Cauchy-Schwarz inequality says more, it also says that |a,b|=aba=λb where λ is a constant.

So if |fg,g¯=fgg¯ then fg=λg¯, where λ is a constant. If g is a canonical parameterization |g|=1 and we get the absolute modulusPlanetmathPlanetmath |λ|=|fg| (which must be constant) and all that remains is to find the phase of λ which must also be constant.

Title proof of estimating theorem of contour integral
Canonical name ProofOfEstimatingTheoremOfContourIntegral
Date of creation 2013-03-22 15:46:02
Last modified on 2013-03-22 15:46:02
Owner cvalente (11260)
Last modified by cvalente (11260)
Numerical id 22
Author cvalente (11260)
Entry type Proof
Classification msc 30E20
Classification msc 30A99