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QUESTION


Let $C_0$ denote the circle $|z-z_0|=R$, taken counterclockwise.
Prove that $$\int_{C_0}{dz\over z-z_0}=2\pi i$$ and
$$\int_{C_0}(z-z_0)^{n-1}ds=0, \ \ (n=\pm 1,\pm 2,\dots). $$



ANSWER


By question 3 , $$\int_{C_0}{dz\over z-z_0}=\int_C {dz\over
z}=\int_0^{2\pi}{iRe^{it}dt\over Re^{it}}=2\pi i.$$ Also.
$$\int_{C_0}(z-z_0)^{n-1}dz=\int_Cz^{n-1}dz=
\int_0^{2\pi}R^{n-1}e^{n-1}Rie^{it}dt=R^{ni}\int_0^{2\pi}e^{nit}dt$$
(using the substitution $z=Re^{it}$). We thus get ${R^n\over
n}i[e^{nit}]_0^{2\pi}=0$ if $n\not=0$.




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