From b248bb1e441beeaf6d8457983cc7542e46ab4382 Mon Sep 17 00:00:00 2001 From: Trance-0 <60459821+Trance-0@users.noreply.github.com> Date: Tue, 30 Sep 2025 19:57:27 -0500 Subject: [PATCH] updates --- .../Math401/Extending_thesis/Math401_S4.md | 47 ++++++++++++++++++- content/Math416/Math416_L1.md | 15 ++++-- content/Math416/Math416_L18.md | 24 ++++++---- content/Math416/Math416_L4.md | 20 ++++---- content/Math416/Math416_L5.md | 13 ++--- content/Math416/Math416_L6.md | 24 ++++++---- content/Math416/Math416_L7.md | 15 +++--- content/Math416/Math416_L8.md | 10 ++-- 8 files changed, 121 insertions(+), 47 deletions(-) diff --git a/content/Math401/Extending_thesis/Math401_S4.md b/content/Math401/Extending_thesis/Math401_S4.md index 793338f..9579beb 100644 --- a/content/Math401/Extending_thesis/Math401_S4.md +++ b/content/Math401/Extending_thesis/Math401_S4.md @@ -87,7 +87,52 @@ $$ P_s(z)=\{v\in \mathbb{C}^d: |v_k-z_k|1$, we can use the same argument to show that + +Let $\mathbb{I}_{P_s(z)}(v)=\begin{cases}1 & v\in P_s(z) \\0 & v\notin P_s(z)\end{cases}$ be the indicator function of $P_s(z)$. + +$$ +\begin{aligned} +F(z)&=(\pi s^2)^{-d}\int_{U}\mathbb{I}_{P_s(z)}(v)\frac{1}{\alpha(v)}F(v)\alpha(v) d\mu(v)\\ +&=(\pi s^2)^{-d}\langle \mathbb{I}_{P_s(z)}\frac{1}{\alpha},F\rangle_{L^2(U,\alpha)} +\end{aligned} +$$ + +By definition of inner product. + +So $\|F(z)\|^2\leq (\pi s^2)^{-2d}\|\mathbb{I}_{P_s(z)}\frac{1}{\alpha}\|^2_{L^2(U,\alpha)} \|F\|^2_{L^2(U,\alpha)}$. + +All the terms are bounded and finite. + +For part 2, we need to show that $\forall z\in U$, we can find a neighborhood $V$ of $z$ and a constant $d_z$ such that + +$$ +|F(z)|^2\leq d_z \|F\|^2_{L^2(U,\alpha)} +$$ diff --git a/content/Math416/Math416_L1.md b/content/Math416/Math416_L1.md index 13a2c16..806c5bc 100644 --- a/content/Math416/Math416_L1.md +++ b/content/Math416/Math416_L1.md @@ -81,7 +81,8 @@ $$ |z_1+z_2|\leq |z_1|+|z_2| $$ -Proof: +
+Proof Geometrically, the triangle inequality states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side. @@ -97,6 +98,8 @@ $$ \end{aligned} $$ +
+ Suppose $2(|z_1||z_2|-|z_1z_2|)=0$, and $\overline{z_1}z_2$ is a non-negative real number $c$, then $|z_1||z_2|=|z_1z_2|$... > What is the use of this? @@ -113,7 +116,8 @@ $$ The sum of the squares of the lengths of the diagonals of a parallelogram equals the sum of the squares of the lengths of the sides. -Proof: +
+Proof Let $z_1,z_2$ be two complex numbers representing the two sides of the parallelogram, then the sum of the squares of the lengths of the diagonals of the parallelogram is $|z_1-z_2|^2+|z_1+z_2|^2$, and the sum of the squares of the lengths of the sides is $2|z_1|^2+2|z_2|^2$. @@ -125,7 +129,7 @@ $$ \end{aligned} $$ -QED +
#### Definition 1.9 @@ -143,12 +147,15 @@ $$ z^n=r^n\text{cis}(n\theta) $$ -Proof: +
+Proof For $n=0$, $z^0=1=1\text{cis}(0)$. For $n=-1$, $z^{-1}=\frac{1}{z}=\frac{1}{r}\text{cis}(-\theta)=\frac{1}{r}(cos(-\theta)+i\sin(-\theta))$. +
+ Application: $$ diff --git a/content/Math416/Math416_L18.md b/content/Math416/Math416_L18.md index 712c656..3fb3c90 100644 --- a/content/Math416/Math416_L18.md +++ b/content/Math416/Math416_L18.md @@ -26,21 +26,27 @@ $$ \int_{C(z_0,r)} f(z) dz = \sum_{n=-\infty}^{\infty} c_n \int_{C(z_0,r)} (z-z_0)^n dz $$ -> $$ -\int_{C(z_0,r)} (z-z_0)^n dz = \begin{cases} - 2\pi i, & n=-1 \\ - 0, & n\neq -1 -\end{cases}$$ -> Proof: -> $\gamma(t)=z_0+re^{it}, t\in[0,2\pi]$ -> $$\begin{aligned} +
+Additional Proof + +$$ +\int_{C(z_0,r)} (z-z_0)^n dz = \begin{cases} 2\pi i, & n=-1 \\0, & n\neq -1\end{cases} +$$ + +Proof: + +$\gamma(t)=z_0+re^{it}, t\in[0,2\pi]$ +$$ +\begin{aligned} \int_{C(z_0,r)} (z-z_0)^n dz &= \int_0^{2\pi} (z_0+re^{it}-z_0)^n ire^{it} dt \\ &= ir^{n+1} \int_0^{2\pi} e^{i(n+1)t} dt \\ &= \begin{cases} 2\pi i, & n=-1 \\ \int_0^{2\pi} e^{i(n+1)t} dt = \frac{1}{i(n+1)}e^{i(n+1)t}\Big|_0^{2\pi} = 0, & n\neq -1 \end{cases} -\end{aligned}$$ +\end{aligned} +$$ +
So, diff --git a/content/Math416/Math416_L4.md b/content/Math416/Math416_L4.md index 204dd4d..edb4c4b 100644 --- a/content/Math416/Math416_L4.md +++ b/content/Math416/Math416_L4.md @@ -44,7 +44,8 @@ $$ > Looks like the chain rule. -Proof: +
+Proof We want to show that @@ -87,7 +88,7 @@ $$ \end{aligned} $$ -QED +
#### Definition 2.12 (Conformal function) @@ -111,7 +112,8 @@ Suppose $f$ is real differentiable, let $a=\frac{\partial f}{\partial z}(z_0)$, Let $\gamma(t_0)=z_0$. Then $(f\circ \gamma)'(t_0)=a\gamma'(t_0)+b\overline{\gamma'(t_0)}$. -Proof: +
+Proof $f=u+iv$, $u,v$ are real differentiable. @@ -144,7 +146,7 @@ $$ \end{aligned} $$ -QED +
#### Theorem of differentiability @@ -152,7 +154,8 @@ Let $f:G\to \mathbb{C}$ be a function defined on an open set $G\subset \mathbb{C Then, $f$ is conformal at every point $z_0\in G$ if and only if $f$ is holomorphic at $z_0$ and $f'(z_0)\neq 0$. -Proof: +
+Proof We prove the equivalence in two parts. @@ -193,7 +196,7 @@ $$ $$ for any differentiable curve $\gamma$ through $z_0$, then the effect of $f$ near $z_0$ is exactly given by multiplication by $f'(z_0)$. Since multiplication by a nonzero complex number is a similarity transformation, $f$ is conformal at $z_0$. -QED +
### Harmonic function @@ -211,7 +214,8 @@ $$ Let $f=u+iv$ be holomorphic function on domain $\Omega\subset \mathbb{C}$. Then $u$ and $v$ are harmonic functions on $\Omega$. -Proof: +
+Proof $$ \Delta u=\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0. @@ -229,7 +233,7 @@ $$ \Delta u=\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=\frac{\partial^2 v}{\partial x\partial y}-\frac{\partial^2 v}{\partial y\partial x}=0. $$ -QED +
If $v$ is such that $f=u+iv$ is holomorphic on $\Omega$, then $v$ is called harmonic conjugate of $u$ on $\Omega$. diff --git a/content/Math416/Math416_L5.md b/content/Math416/Math416_L5.md index 7c06600..f659ad6 100644 --- a/content/Math416/Math416_L5.md +++ b/content/Math416/Math416_L5.md @@ -14,7 +14,7 @@ Df(x+iy)=\begin{pmatrix} \end{pmatrix} $$ -So +So, $$ \begin{aligned} @@ -53,7 +53,6 @@ $$ ## Chapter 3: Linear fractional Transformations - Let $a,b,c,d$ be complex numbers. such that $ad-bc\neq 0$. The linear fractional transformation is defined as @@ -185,7 +184,8 @@ So the kernel of $F$ is the set of matrices that represent the identity transfor If $\phi$ is a non-constant linear fractional transformation, then $\phi$ is conformal. -Proof: +
+Proof Know that $\phi_0\circ\phi(z)=z$, @@ -197,13 +197,14 @@ $\phi:\mathbb{C}\cup\{\infty\}\to\mathbb{C}\cup\{\infty\}$ which gives $\phi(\in So, $\phi$ is conformal. -QED +
#### Proposition 3.4 of Fixed points Any non-constant linear fractional transformation except the identity transformation has 1 or 2 fixed points. -Proof: +
+Proof Let $\phi(z)=\frac{az+b}{cz+d}$. @@ -221,7 +222,7 @@ Such solutions are $z=\frac{-(d-a)\pm\sqrt{(d-a)^2+4bc}}{2c}$. So, $\phi$ has 1 or 2 fixed points. -QED +
#### Proposition 3.5 of triple transitivity diff --git a/content/Math416/Math416_L6.md b/content/Math416/Math416_L6.md index b6e18aa..1c7ce5e 100644 --- a/content/Math416/Math416_L6.md +++ b/content/Math416/Math416_L6.md @@ -36,7 +36,8 @@ when $\alpha=0$, it is a line. when $\alpha\neq 0$, it is a circle. -Proof: +
+Proof Let $w=u+iv=\frac{1}{z}$, so $\frac{1}{w}=\frac{u}{u^2+v^2}-i\frac{v}{u^2+v^2}$. @@ -48,7 +49,7 @@ $$ Which is in the form of circle equation. -QED +
## Chapter 4 Elementary functions @@ -83,7 +84,8 @@ $$ $e^z$ is holomorphic on $\mathbb{C}$. -Proof: +
+Proof $$ \begin{aligned} @@ -93,19 +95,20 @@ $$ \end{aligned} $$ -QED +
#### Theorem 4.4 $e^z$ is periodic $e^z$ is periodic with period $2\pi i$. -Proof: +
+Proof $$ e^{z+2\pi i}=e^z e^{2\pi i}=e^z\cdot 1=e^z $$ -QED +
#### Theorem 4.5 $e^z$ as a map @@ -185,13 +188,14 @@ A branch of $\log(z)$ in $G$ is a continuous function $\beta$, such that $e^{\be Note: $G$ has a branch of $\arg(z)$ if and only if it has a branch of $\log(z)$. -Proof: +
+Proof Suppose there exists $\alpha(z)$ such that $\forall z\in G$, $\alpha(z)\in G$, then $l(z)=\ln|z|+i\alpha(z)$ is a branch of $\log(z)$. Suppose there exists $l(z)$ such that $\forall z\in G$, $l(z)\in G$, then $\alpha(z)=Im(z)$ is a branch of $\arg(z)$. -QED +
If $G=\mathbb{C}\setminus\{0\}$, then not branch of $\arg(z)$ exists. @@ -222,7 +226,8 @@ for some $k\in\mathbb{Z}$. $\log(z)$ is holomorphic on $\mathbb{C}\setminus\{0\}$. -Proof: +
+Proof (continue on next lecture) Method 1: Use polar coordinates. (See in homework) @@ -238,3 +243,4 @@ $$ $$ Continue next time. +
\ No newline at end of file diff --git a/content/Math416/Math416_L7.md b/content/Math416/Math416_L7.md index fd1b151..578bc71 100644 --- a/content/Math416/Math416_L7.md +++ b/content/Math416/Math416_L7.md @@ -26,7 +26,8 @@ A branch of logarithm is a continuous function $f$ on a domain $D$ such that $e^ $\log(z)$ is holomorphic on $\mathbb{C}\setminus\{0\}$. -Proof: +
+Proof We proved that $\frac{\partial}{\partial\overline{z}}e^{z}=0$ on $\mathbb{C}\setminus\{0\}$. @@ -36,7 +37,7 @@ Since $\frac{d}{dz}e^{z}=e^{z}$, we know that $e^{z}$ is conformal, so any branc Since $\exp(\log(z))=z$, we know that $\log(z)$ is the inverse of $\exp(z)$, so $\frac{d}{dz}\log(z)=\frac{1}{e^{\log(z)}}=\frac{1}{z}$. -QED +
We call $\frac{f'}{f}$ the logarithmic derivative of $f$. @@ -78,7 +79,8 @@ If $|c|<1$, then $\lim_{N\to\infty}\sum_{n=0}^{N}c^n=\frac{1}{1-c}$. otherwise, the series diverges. -Proof: +
+Proof The geometric series converges if $\frac{c^{N+1}}{1-c}$ converges. @@ -90,7 +92,7 @@ If $|c|<1$, then $\lim_{N\to\infty}c^{N+1}=0$, so $\lim_{N\to\infty}(1-c)(1+c+c^ If $|c|\geq 1$, then $c^{N+1}$ does not converge to 0, so the series diverges. -QED +
#### Theorem 5.4 (Triangle Inequality for Series) @@ -146,7 +148,8 @@ For every power series, there exists a radius of convergence $r$ such that the s And it diverges pointwise outside $B_r(z_0)$. -Proof: +
+Proof Without loss of generality, we can assume that $z_0=0$. @@ -166,7 +169,7 @@ So the series converges absolutely and uniformly on $\overline{B_r(0)}$. If $|z| > r$, then $|c_n z^n|$ does not tend to zero, and the series diverges. -QED +
We denote this $r$ captialized by te radius of convergence diff --git a/content/Math416/Math416_L8.md b/content/Math416/Math416_L8.md index 3152e95..72ed268 100644 --- a/content/Math416/Math416_L8.md +++ b/content/Math416/Math416_L8.md @@ -67,7 +67,8 @@ $$ \frac{1}{R} = \limsup_{n\to\infty} |a_n|^{1/n} $$ -Proof: +
+Proof Suppose $(b_n)^{\infty}_{n=0}$ is a sequence of real numbers such that $\lim_{n\to\infty} b_n$ may nor may not exists by $(-1)^n(1-\frac{1}{n})$. @@ -111,7 +112,7 @@ So $\sum_{n=0}^{\infty} a_n (z - z_0)^n$ does not converge at $z$ if $|z|> \frac So $R=\frac{1}{\rho}$. -QED +
_What if $|z-z_0|=R$?_ @@ -135,7 +136,8 @@ Suppose $\sum_{n=0}^{\infty} a_n (z - z_0)^n$ has a positive radius of convergen > Here below is the proof on book, which will be covered in next lecture. -Proof: +
+Proof Without loss of generality, assume $z_0=0$. Let $R$ be the radius of convergence for the two power series: $\sum_{n=0}^{\infty} a_n z^n$ and $\sum_{n=1}^{\infty} n a_n z ^{n-1}$. The two power series have the same radius of convergence $|R|$. @@ -179,4 +181,4 @@ So $\left|\frac{f(z)-f(z_1)}{z-z_1}-g(z_1)\right|\leq M|z-z_1|$ for $|z|<\rho$. So $\lim_{z\to z_1}\frac{f(z)-f(z_1)}{z-z_1}=g(z_1)$. -QED +