Using Rodrigues' formula one can express the Legendre polynomial $P_n(x)$ as a contour integral round a simple loop $K$ which has $z$ inside, as
$$P_n(z)=\frac{1}{2\pi i}\oint_K\frac{(Z^2-1)^n}{2^n(Z-z)^{n+1}}dZ$$
By taking $K$ to be any circle centred at $z$ and integrating, we obtain a formula for $P_n(z)$ which is different for each circle. Why are they not all the same? Shouldn't they all be the same?
For example to get the Legendre polynomials one has to choose a circle of radius $\sqrt{|z^2-1|}$, centred at $z$.