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Math416 Lecture 13

Review on Cauchy's Theorem

Cauchy's Theorem states that if a function is holomorphic (complex differentiable) on a simply connected domain, then the integral of that function over any closed contour within that domain is zero.

Last lecture we proved the case for convex regions.

Cauchy's Formula for a Circle

Let C be a counterclockwise oriented circle, and let f be a holomorphic

function defined in an open set containing C and its interior. Then,


f(z_0)=\frac{1}{2\pi i}\int_C\frac{f(z)}{z-z_0}dz

for all points z in the interior of C.

New materials

Mean value property

Theorem 7.6: Mean value property

Special case: Suppose f is holomorphic on some \mathbb{D}(z_0,R)\subset \mathbb{C}, by cauchy's formula,


f(z_0)=\frac{1}{2\pi i}\int_{C_r}\frac{f(z)}{z-z_0}dz

Parameterizing C_r, we get \gamma(t)=z_0+re^{it}, 0\leq t\leq 2\pi


\int f(z)dz=\int f(\gamma) \gamma'(t) d t

So,


f(z_0)=\frac{1}{2\pi i}\int_0^{2\pi}\frac{f(z_0+re^{it})}{re^{it}} ire^{it} dt=\frac{1}{2\pi}\int_{0}^{2\pi} f(z_0+re^{it}) dt

This concludes the mean value property for the holomorphic function

If f is holomorphic, f(z_0) is the mean value of f on any circle centered at z_0

Area representation of mean value property

Area of f on \mathbb{D}(z_0,r)


\frac{1}{\pi r^2}\int_{0}^{2\pi}\int_0^r f(z_0+re^{it})

/Track lost/

Cauchy Integral

Definition 7.7

Let \gamma:[a,b]\to \mathbb{C} be piecewise \mathbb{C}^1, let \phi be condition on \gamma. Then the Cauchy interval of \phi along \gamma is


F(z)=\int_{\gamma}\frac{\phi(\zeta)}{\zeta-z}d \zeta

Theorem

Suppose F(z)=\int_{\gamma}\frac{\phi(z)}{\zeta-z}d z. Then F has a local power series representation at all points z_0 not in \gamma.

Proof:

Let R=B(z_0,\gamma)>0, let z\in \mathbb{D}(z_0,R)


\frac{1}{\zeta-z}=\frac{1}{(\zeta-z_0)-(z-z_0)}=\frac{1}{1-z_0}\frac{1}{1-\frac{z-z_0}{\zeta-z_0}}

Since |z-z_0|<R and |\zeta-z_0|>R, |\frac{z-z_0}{\zeta-z_0}|<1.

Converting it to geometric series


\frac{1}{1-z_0}\frac{1}{1-\frac{z-z_0}{\zeta-z_0}}=\sum_{n=0}^\infty \left(\frac{z-z_0}{\zeta-z_0}\right)^n

So,


\begin{aligned}
F(z)&=\int_\gamma \frac{\phi(\zeta)}{\zeta - z} d\zeta\\
&=\int_\gamma \frac{\phi(\zeta)}{z-z_0} \sum_{n=0}^\infty \left(\frac{z-z_0}{\zeta-z_0}\right)^n dz\\
&=\sum_{n=0}^\infty (z-z_0)^n \int_\gamma \frac{\phi(\zeta)}{(\zeta-z_0)^{n+1}} 
\end{aligned}

Which gives us an power series representation if we set a_n=\int_\gamma \frac{\phi(\zeta)}{(\zeta-z_0)^{n+1}}

QED

Corollary 7.7

Suppose F(z)=\int_\gamma \frac{\phi(\zeta)}{\zeta-z_0} dz,

Then,


f^{(n)}(z)=n!\int_\gamma \frac{\phi(z)}{(\zeta-z_0)^{n+1}} d\zeta

where z\in \mathbb{C}\setminus \gamma.

Combine with Cauchy integral formula:

If f is in O(\Omega), then \forall z\in \mathbb{D}(z_0,r).


f(z)=\frac{1}{2\pi i}\int_{C_r}\frac{f(\zeta)}{\zeta-z} d\zeta

We have proved that If f\in O(\Omega), then f is locally given by a convergent power series

power series has radius of convergence at z_0 that is \geq dist(z_0,boundary \Omega)

Liouville's Theorem

Definition 7.11

A function that is holomorphic in all of \mathbb{C} is called an entire function.

Theorem 7.11 Liouville's Theorem

Any bounded entire function is constant.

Basic Estimate of integral
\left|\int_\gamma f(z) dz\right|\leq L(\gamma) \max\left|f(z)\right|

Since,


f'(z)=\frac{1}{2\pi i} \int_{C_r} \frac{f(z)}{(\zeta-z)^2} dz

So the modulus of the integral is bounded by


\frac{1}{2\pi} |M|\cdot \frac{1}{R^2}=2\pi R\cdot M \frac{1}{R^2}=\frac{M}{R}

Fundamental Theorem of Algebra

Theorem 7.12 Fundamental Theorem of ALgebra

Every nonconstant polynomial with complex coefficients can be factored over \mathbb{C} into linear factors.

Corollary

For every polynomial with complex coefficients.


p(z)=c\prod_{j=i}^n(z-z_0)^{t_j}

where the degree of polynomial is \sum_{j=0}^n t_j