- #1
sikrut
- 49
- 1
(a) Suppose [itex]\kappa[/itex] is a clockwise circle of radius [itex]R[/itex] centered at a complex number [itex]\mathcal{z}[/itex]0. Evaluate: [tex]K_m := \oint_{\kappa}{dz(z-z_0)^m} [/tex]
for any integer [itex] m = 0, \pm{1},\pm{2}, ,... [/itex]Show that
[itex] K_m = -2\pi i[/itex] if [itex] m = -2;[/itex] else :[itex] K_m = 0 [/itex] if [itex] m = 0, \pm{1}, \pm{2}, \pm{3},...[/itex]
Note the minus sign here: [itex]\kappa[/itex] 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 [itex]K_m[/itex] using
[itex]z(\tau) = c + re^{i\tau} , \tau \in [a,b][/itex] with [itex] a \equiv {\theta_a}[/itex] and [itex] b \equiv \theta_b,[/itex] if [itex] \theta_a < \theta_b, [/itex]
But I'm just stuck on how to set this up at this point. Any ideas?
for any integer [itex] m = 0, \pm{1},\pm{2}, ,... [/itex]Show that
[itex] K_m = -2\pi i[/itex] if [itex] m = -2;[/itex] else :[itex] K_m = 0 [/itex] if [itex] m = 0, \pm{1}, \pm{2}, \pm{3},...[/itex]
Note the minus sign here: [itex]\kappa[/itex] 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 [itex]K_m[/itex] using
[itex]z(\tau) = c + re^{i\tau} , \tau \in [a,b][/itex] with [itex] a \equiv {\theta_a}[/itex] and [itex] b \equiv \theta_b,[/itex] if [itex] \theta_a < \theta_b, [/itex]
But I'm just stuck on how to set this up at this point. Any ideas?