# 記號約定

(1)
\begin{align} z = x + i y\, \end{align}

(2)

(3)
\begin{align} z = x + iy = |z|\left(\cos\theta + i\sin\theta\right) = |z|e^{i\theta}\, \end{align}

(4)
\begin{align} |z| = \sqrt{x^2+y^2}; \quad \theta = \arg(z) = -i\log\frac{z}{|z|}.\, \end{align}

(5)
\begin{align} z = x + iy;\qquad f(z) = w = u + iv,\, \end{align}

# 複變函數

(6)
\begin{align} z = x + iy\, \end{align}
(7)
\begin{align} w = f(z) = u(x,y) + iv(x,y)\, \end{align}

(8)
\begin{align} u = u(x,y)\, \end{align}
(9)
\begin{align} v = v(x,y),\, \end{align}

# 全純函數

(10)
\begin{align} { \partial u \over \partial x } = { \partial v \over \partial y } \end{align}

(11)
\begin{align} { \partial u \over \partial y } = -{ \partial v \over \partial x } . \end{align}

# 柯西積分定理

(12)
\begin{align} f(a) = {1 \over 2\pi i} \oint_C {f(z) \over z-a}\, dz \end{align}

(13)
\begin{align} \oint_\gamma f(z)\,dz = 0. \end{align}

U 是單連通的條件，意味著 U 沒有“洞”，例如任何一個開圓盤 U={ z: |z-z0| < r} 都符合條件，這個條件是很重要的，考慮以下路徑

(14)
\begin{align} \gamma(t) = e^{it} \quad t \in \left[0,2\pi\right] \end{align}

(15)
\begin{align} \oint_\gamma \frac{1}{z}\,dz = \int_0^{2\pi} { ie^{it} \over e^{it} }\,dt= \int_0^{2\pi}i\,dt = 2\pi i \end{align}

(16)
\begin{align} \int_\gamma f(z)\,dz=F(b)-F(a). \end{align}

# 柯西積分公式

(17)
\begin{align} f(a) = {1 \over 2\pi i} \oint_C {f(z) \over z-a}\, dz \end{align}

# 複變函數的級數展開

(18)
\begin{align} f(z) = \sum_{n=0}^\infty a_n (z-z_0)^n = a_0 + a_1 (z-z_0) + a_2 (z-z_0)^2 + a_3 (z-z_0)^3 + \cdots \end{align}

# 洛朗級數

(19)
\begin{align} f(z)=\sum_{n=-\infty}^\infty a_n(z-c)^n \end{align}
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