127 lines
3.3 KiB
Markdown
127 lines
3.3 KiB
Markdown
# Math 416 Final Review
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Story after Cauchy's theorem
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## Chapter 7: Cauchy's Theorem
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### Existence of harmonic conjugate
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Suppose $f=u+iv$ is holomorphic on a domain $U\subset\mathbb{C}$. Then $u=\Re f$ is harmonic on $U$. That is $\Delta u=\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0$.
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Moreover, there exists $g\in O(U)$ such that $g$ is unique up to an additive imaginary constant.
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> Example:
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> Find a harmonic conjugate of $u(x,y)=\log|\frac{z}{z-1}|$
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> Note that $\log(\frac{z}{z-1})=\log \left|\frac{z}{z-1}\right|+i(\arg(z)-\arg(z-1))$ is harmonic on $\mathbb{C}\setminus\{1\}$.
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>
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> So the harmonic conjugate of $u(x,y)=\log|\frac{z}{z-1}|$ is $v(x,y)=\arg(z)-\arg(z-1)+C$ where $C$ is a constant.
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>
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> Note that the harmonic conjugate may exist locally but not globally. (e.g. $u(x,y)=\log|z(z-1)|$ has a local harmonic conjugate $i(\arg(z)+\arg(z-1)+C)$ but this is not a well defined function since $\arg(z)+\arg(z-1)$ is not single-valued.)
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### Corollary for harmonic functions
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#### Theorem 7.19
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Harmonic function are infinitely differentiable.
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#### Theorem 7.20
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Mean value property of harmonic functions.
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Let $u$ be harmonic on an open set of $\Omega$, then
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$$u(z_0)=\frac{1}{2\pi}\int_0^{2\pi} u(z_0+re^{i\theta}) d\theta$$
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for any $z_0\in\Omega$ and $r>0$ such that $D(z_0,r)\subset\Omega$.
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#### Theorem 7.21
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Identity theorem for harmonic functions.
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Let $u$ be harmonic on a domain $\Omega$. If $u=0$ on some open set $G\subset\Omega$, then $u\equiv 0$ on $\Omega$.
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#### Theorem 7.22
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Maximum principle for harmonic functions.
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Let $u$ be a non-constant real-valued harmonic function on a domain $\Omega$. Then $|u|$ does not attain a maximum value in $\Omega$.
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## Chapter 8: Laurent Series and Isolated Singularities
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### Laurent Series
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Laurent series is a generalization of Taylor series.
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Laurent series is a power series of the form
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$$f(z)=\sum_{n=-\infty}^{\infty} a_n (z-z_0)^n$$
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where
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$$
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a_k=\frac{1}{2\pi i}\int_{C_r}\frac{f(z)}{(z-z_0)^{k+1}}dz
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$$
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The series converges on an annulus $R_1<|z-z_0|<R_2$.
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where $R_1=\limsup_{n\to\infty} |a_{-n}|^{1/n}$ and $R_2=\frac{1}{\limsup_{n\to\infty} |a_n|^{1/n}}$.
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### Cauchy's Formula for an Annulus
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Let $f$ be holomorphic on an annulus $R_1<r_1<|z-z_0|<r_2<R_2$. And $w\in A(z_0;R_1,R_2)$. Find the Annulus $w\in A(z_0;r_1,r_2)$
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Then
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$$
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f(w)=\frac{1}{2\pi i}\int_{C_{r_1}}\frac{f(z)}{z-w}dz-\frac{1}{2\pi i}\int_{C_{r_2}}\frac{f(z)}{z-w}dz
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$$
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### Isolated singularities
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Let $f$ be holomorphic on $0<|z-z_0|<R$ (The punctured disk of radius $R$ centered at $z_0$). If there exists a Laurent series of $f$ convergent on $0<|z-z_0|<R$, then $z_0$ is called an isolated singularity of $f$.
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The series $f(z)=\sum_{n=-\infty}^{-1}a_n(z-z_0)^n$ is called the principle part of Laurent series of $f$ at $z_0$.
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#### Removable singularities
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If the principle part of Laurent series of $f$ at $z_0$ is zero, then $z_0$ is called a removable singularity of $f$.
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> Example:
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> $f(z)=\frac{e^z-1}{z^2}$ has a removable singularity at $z=0$.
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> The Laurent series of $f$ at $z=0$ can be found using the Taylor series of $e^z-1$ at $z=0$.
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> $$e^z-1=z+\frac{z^2}{2!}+\frac{z^3}{3!}+\cdots$$
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> So the Laurent series of $f$ at $z=0$ is
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> $$f(z)=\frac{1}{z^2}+\frac{1}{z}+\sum_{n=0}^{\infty}\frac{z^n}{n!}$$
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> The principle part is zero, so $z=0$ is a removable singularity.
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#### Poles
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