计算下列积分(1) int_(|z|=2) (2z^2 - z + 1)/(z-1) dz(2) int_(|z|=1) (e^z)/(z^100) dz(3) int_(|z|=2) (sin z)/((z-frac(pi){2))^2} dz
计算下列积分
(1) $\int_{|z|=2} \frac{2z^2 - z + 1}{z-1} dz$
(2) $\int_{|z|=1} \frac{e^z}{z^{100}} dz$
(3) $\int_{|z|=2} \frac{\sin z}{(z-\frac{\pi}{2})^2} dz$
题目解答
答案
(1) 由柯西积分公式，$f(z) = 2z^2 - z + 1$，$a = 1$，得
$\oint_{|z|=2} \frac{2z^2 - z + 1}{z - 1} \, dz = 2\pi i f(1) = 2\pi i \times 2 = 4\pi i$
答案： $4\pi i$
(2) 由柯西积分公式一般形式，$f(z) = e^z$，$a = 0$，$n = 99$，得
$\oint_{|z|=1} \frac{e^z}{z^{100}} \, dz = \frac{2\pi i}{99!} f^{(99)}(0) = \frac{2\pi i}{99!} \times 1 = \frac{2\pi i}{99!}$
答案： $\frac{2\pi i}{99!}$
(3) 由柯西积分公式一般形式，$f(z) = \sin z$，$a = \frac{\pi}{2}$，$n = 1$，得
$\oint_{|z|=2} \frac{\sin z}{(z - \frac{\pi}{2})^2} \, dz = 2\pi i f'\left(\frac{\pi}{2}\right) = 2\pi i \times 0 = 0$
答案： $0$
$\boxed{\begin{array}{ccc}\text{(1) } 4\pi i \\\text{(2) } \frac{2\pi i}{99!} \\\text{(3) } 0 \\\end{array}}$