Let $f$ be a continuous function on $\mathbb{C}$ and assume that $\lim_{z\to \infty} zf(z) = \lambda.$ Let us note for all natural $n$ $$C_n = \{z \in \mathbb{C} : |z|=n\}.$$ Then, a usual fact of complex analysis states that $$\lim_{n \to +\infty} \int_{C_n} f(z) dz = 2i\pi\lambda.$$ This is easily proven with a parametrization of the circles. Actually I'm interested by a possible generalization of this fact. Suppose that we have $f$ with same assumptions and for each $n$ a $1$-singular chain $\gamma_n$ supported by $C_n$. Can we say that $$\lim_{n \to +\infty} \int_{\gamma_n} f(z) dz = 2i\pi\lambda\quad ?$$ I tried to adapt the previous proof but it seems complicated since the number of "pieces" of $\gamma_n$ depends on $n$. Any help will be greatly appreciated.
Edit I'm now pretty sure the limit can't be $2i\pi\lambda$ but the existence of the limit would already be very helpful.
