- #1

PhDeezNutz

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- Homework Statement
- Prove that ## \left| \int_C f \left(z \right) \, dz \right| \leq \left|f \right|_{max} \cdot L## where ##\left|f \right|_{max}## is the maximum value o ##\left|f(z) \right|## on the contour and ##L## is the arc length of the contour

- Relevant Equations
- Any complex valued function is of the form ##f \left(z\right) = u \left(x,y\right) + i v\left(x,y\right)##

I think throughout this I need to use the triangle inequality repeatedly

##\left|\vec{a} + \vec{b} \right| \leq \left| \vec{a}\right| + \left|\vec{b} \right|##

Also the Cauchy Schwarz Inequality

##\left| \vec{a} \cdot \vec{b} \right| \leq \left| \vec{a} \right| \left| \vec{b} \right|##

Just for good measure

##\left| f \left(z\right) \right| = \sqrt{u^2 + v^2}##

Here is my attempt (Note:

## \left| \int_{C} f \left( z \right) \, dz \right| \leq \left| \int_C udx -vdy +ivdx +iudy \right|##

##= \left| \int_{C} \left( u+iv, -v +iu \right) \cdot \left(dx, dy \right) \right| ##

Here I am going to surround the above expression with another set of absolute value bars

##\leq \left| \left| \int_{C} \left( u+iv, -v +iu \right) \cdot \left(dx, dy \right) \right| \right| ##

Appealing to Cauchy-Schwarz

##\leq \left| \left| \int_{C} \left| \left( u+iv, -v +iu \right)\right| \left| \left(dx, dy \right)\right| \right| \right| ##

taking the dot product of the newly defined vector field with itself and noting that cross terms vanish

## = \left| \left| \int_{C} \sqrt{u^2 + v^2}\left| \left(dx,dy\right)\right| \right| \right| \leq \left| \left(\sqrt{u^2 + v^2} \right)_{max} \int_{C} \left| \left(dx,dy\right) \right| \right|##

## \leq f_{max} L##

## \left| \int_{C} f \left( z \right) \, dz \right| \leq \left| \int_C udx -vdy +ivdx +iudy \right|##

##= \left| \int_{C} \left( u+iv, -v +iu \right) \cdot \left(dx, dy \right) \right| ##

Here I am going to surround the above expression with another set of absolute value bars

##\leq \left| \left| \int_{C} \left( u+iv, -v +iu \right) \cdot \left(dx, dy \right) \right| \right| ##

Appealing to Cauchy-Schwarz

##\leq \left| \left| \int_{C} \left| \left( u+iv, -v +iu \right)\right| \left| \left(dx, dy \right)\right| \right| \right| ##

taking the dot product of the newly defined vector field with itself and noting that cross terms vanish

## = \left| \left| \int_{C} \sqrt{u^2 + v^2}\left| \left(dx,dy\right)\right| \right| \right| \leq \left| \left(\sqrt{u^2 + v^2} \right)_{max} \int_{C} \left| \left(dx,dy\right) \right| \right|##

## \leq f_{max} L##

Last edited: