복소수함수론/코시 적분공식4 Cauchy-Goursat Theorem Thm 1 Cauchy-Goursat Theorem Let $\Omega$ be an area, $R \subset \Omega$ be a rectangle and $f : \Omega \to \mathbb{C}$ be analytic function. Then$\displaystyle \int_{\partial R} f(z) \; \mathrm{d}z = 0$ 더보기 Divide a rectangle $R$ into four equal parts by bisecting each side of $R$, and label these parts as $R^{1}$, $R^{2}$, $R^{3}$, and $R^{4}$. By the cancellation of paths, $\displaystyl.. 2024. 9. 9. The Fundamental Theorem of Calculus Def 1Let $\Omega$ be an area and $F : \Omega \to \mathbb{C}$ be differentiable on $\Omega$If $F' = f$, $F$ is called primitive function of $f$ Thm 1Let $\Omega$ be an area and $f : \Omega \to \mathbb{C}$ be continuous on $\Omega$Let $F$ be a primitive function of $f$ and $\Gamma : [a, b] \to \Omega$ be a curve. Then$\displaystyle \int_{\Gamma} f(z) \; \mathrm{d}z = F(\Gamma (b)) - F(\Gamm.. 2024. 9. 9. Line integral of complex function Def 1Let $\Omega$ be an area and $\Gamma : [a, b] \to \Omega$ . 1. If both the real and imaginary parts of $\Gamma$ are $C^{1}$-function, it is called a $C^{1}$-curve.2. A curve that divides the interval $[a, b]$ into a finite number of subintervals, where it is a $C^{1}$-curve on each subinterval and continuous over the entire interval $[a, b]$, is called a piecewise $C^{1}$-curve. # From now.. 2024. 9. 7. Integral with complex function Def 1Let $f : \mathbb{R} \to \mathbb{C}$ by$f(t) = u(t) + i v(t)$In this case, if both $u$ and $v$ are integrable, then $f$ is said to be integrable.The same applies to differentiation. We define$\displaystyle \int f \; \mathrm{d}t = \int u \; \mathrm{d}t + i \int v \; \mathrm{d}t$$\displaystyle f' = u' + i v'$ Prop 1Let $f : \mathbb{R} \to \mathbb{C}$ . For $\alpha \in \mathbb{C}$, $\displays.. 2024. 6. 6. 이전 1 다음