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QUESTION


Let $C$ and $C_0$ denote the circles $z=Re^{it}$,  $(0\le t\le
2\pi)$ and $z=z_0+Re^{it}$,

 $(0\le t \le2\pi)$. Draw pictures of these circles and  use
these parametric representations \indent to show that
$$\int_C{f(z)dz}=\int_{C_0}{f(z-z_0)dz}.$$



ANSWER


DIAGRAM

By letting $z=z_0+Re^{it}$ we get
$\int_{C_0}f(z-z_0)dz=\int_0^{2\pi}if(Re^{it})Re^{it}dt$. By
putting $z=Re^{it}$ we get $\int_C
f(z)dz=\int_0^{2\pi}if(Re^{it})Re^{it}dt$. Hence $\int_C
f(z)dz=\int_{C_0}f(z-z_0)dz$.




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