sikrut
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(a) Suppose \kappa is a clockwise circle of radius R centered at a complex number \mathcal{z}0. Evaluate: K_m := \oint_{\kappa}{dz(z-z_0)^m}
for any integer m = 0, \pm{1},\pm{2}, ,...Show that
K_m = -2\pi i if m = -2; else :K_m = 0 if m = 0, \pm{1}, \pm{2}, \pm{3},...
Note the minus sign here: \kappa is clockwise.
I am not allowed to use or assume the validity of the residue theorem, but I can use Cauchy's integral theorem without proof.
I was trying to parameterize K_m using
z(\tau) = c + re^{i\tau} , \tau \in [a,b] with a \equiv {\theta_a} and b \equiv \theta_b, if \theta_a < \theta_b,
But I'm just stuck on how to set this up at this point. Any ideas?
for any integer m = 0, \pm{1},\pm{2}, ,...Show that
K_m = -2\pi i if m = -2; else :K_m = 0 if m = 0, \pm{1}, \pm{2}, \pm{3},...
Note the minus sign here: \kappa is clockwise.
I am not allowed to use or assume the validity of the residue theorem, but I can use Cauchy's integral theorem without proof.
I was trying to parameterize K_m using
z(\tau) = c + re^{i\tau} , \tau \in [a,b] with a \equiv {\theta_a} and b \equiv \theta_b, if \theta_a < \theta_b,
But I'm just stuck on how to set this up at this point. Any ideas?