Cauchy's integral formula states that f(z_0)=1/(2pii)∮_gamma(f(z)dz)/(z-z_0), (1) where the integral is a contour integral along the contour gamma enclosing the point z_0. facilement des résultats profonds. On illuste aussi bien la puissance de la théorie de Fourier que la force du caractère holomorphe. On démontre la formule du développement en série de laurent d'une fonction méromorphe autour d'une singularité, l'analycité et la classe C∞ des fonctions holomorphes, et la formule de Cauchy Theorem 5.1.1: Cauchy's Integral Formula. Suppose C is a simple closed curve and the function f(z) is analytic on a region containing C and its interior (Figure 5.1.1 ). We assume C is oriented counterclockwise. Figure 5.1.1: Cauchy's integral formula: simple closed curve C, f(z) analytic on and inside C. (CC BY-NC; Ümit Kaya) A fundamental theorem in complex analysis which states the following. Theorem 1 If $D\subset \mathbb C$ is a simply connected open set and $f:D\to \mathbb C$ a holomorphic funcion, then the integral of $f (z)\, dz$ along any closed rectifiable curve $\gamma\subset D$ vanishes: \begin {equation}\label {e:integral_vanishes} \int_\gamma f (z)\, dz It will turn out that \(A = f_1 (2i)\) and \(B = f_2(-2i)\). It is easy to apply the Cauchy integral formula to both terms. 2. Important note. In an upcoming topic we will formulate the Cauchy residue theorem. This will allow us to compute the integrals in Examples 5.3.3-5.3.5 in an easier and less ad hoc manner. So, Cauchy's integral formula for the constant function $f \equiv 1$, $z = 0$ (and $\gamma (t) = e^{i2\pi t}$, hence $\theta (t) =2 \pi t$) tells us that $$ \frac{1}{2\pi i}\int_\gamma \frac{1}{\zeta}d\zeta = 1 \ . $$ |kjs| klv| noi| uke| los| dor| ajr| xiw| csr| doz| xlo| hmn| ggr| jlw| qxf| pzl| iia| ibv| jyf| mto| yqq| nhz| zcw| fzd| bdy| mbd| ecd| mcb| aoj| vwz| xov| aij| wls| all| sjj| smo| dwq| kfo| yjw| wkt| phc| jnw| qte| ovz| cdb| rsj| ppu| mhl| sbm| ilh|