Vtyjkyuk

Let $f(z)$ be a meromorphic function with a finite number of poles $a_1,dots,a_m$, where $a_inotinmathbb Zcup0$.

Prove that if there exists a sequence of contours $C_n$ that goes to the point infinity and is such that $$displaystylelim_ntoinftyint_C_nf(z)cot (pi z)dz=0,$$ then $$displaystylesum_n=-infty^infty f(n)=-pisum_k=1^mtextres [f(z)cot(pi z)]_z=a_k$$

From Marsden book:

What does it mean the part that says

Taking limits on both sides...

If you take the limits,

$displaystylelim_Ntoinftysum_n=-N^N f(n)=lim_Ntoinftysumtextres [f(z)picot(pi z)]_z=ndots (*)$

and how do you pass from $(*)$ to $displaystyle-sum_k=1^mtextres [pi f(z)cot(pi z)]_z=a_k$ ?

Could anyone explain please?

Thank you

Taking limits on both sides of the preceding displayed equation for the integral $oint_C_N(picotpi z)f(z)dz$

This means taking the limit $Ntoinfty$ on both sides of
$$
beginalign
oint_C_Nf(z)dz
&=2pi isumtextRes $(picotpi z)f(z)$ at $-N,cdots,+N$ \
&~~~~+2pi isumtextRes $(picotpi z)f(z)$ at singularities of $f$ \
endalign
$$

which is
$$
beginalign
lim_Ntoinftyoint_C_Nf(z)dz
&=lim_Ntoinfty2pi isumtextRes $(picotpi z)f(z)$ at $-N,cdots,+N$ \
&~~~~+lim_Ntoinfty2pi isumtextRes $(picotpi z)f(z)$ at singularities of $f$ \
endalign
$$

and using the fact that $oint_C_N(picotpi z)f(z)dzto0$ as $Ntoinfty$,

Therefore,
$$
beginalign
0
&=lim_Ntoinfty2pi isumtextRes $(picotpi z)f(z)$ at $-N,cdots,+N$ \
&~~~~+lim_Ntoinfty2pi isumtextRes $(picotpi z)f(z)$ at singularities of $f$ qquad(*)\
endalign
$$

Since for an integer $k$ $$textRes $(picotpi z)f(z)$ at $k$=f(k)$$
we can rewrite $(*)$ as
$$
beginalign
0
&=lim_Ntoinfty2pi isum_k=-N^N f(k) \
&~~~~+lim_Ntoinfty2pi isumtextRes $(picotpi z)f(z)$ at singularities of $f$ \
endalign
$$

Furthermore, recognizing that the second sum does not depend on $N$ for large enough $N$, we have

$$beginalign
0
&=lim_Ntoinfty2pi isum_k=-N^N f(k)
+2pi isumtextRes $(picotpi z)f(z)$ at singularities of $f$ \
0
&=lim_Ntoinftysum_k=-N^N f(k)
+sumtextRes $(picotpi z)f(z)$ at singularities of $f$ \
endalign
$$
$$
colorred
lim_Ntoinftysum_k=-N^N f(k)
=-sumtextRes $(picotpi z)f(z)$ at singularities of $f$
$$