I am asking about the part of the zeros and poles of modular functions from A Course in Arithmetic by Jean-Pierre Serre.
My first question is why is it that if $\tilde f$ is meromorphic, then there exists an r>0 such that $\tilde f$ has no zero nor pole for |q| in between 0 and r where q= $e^{2\pi i z}$?
Next my question is when Serre uses the residue theorem. We have that $\int \frac{1}{f}$ is $2\pi i*\sum Res(\frac {1}{f})$ so $\frac {1}{2\pi i}*\int \frac{1}{f}=\sum Res(\frac{1}{f})$
From what Serre writes, the residue is just the order of f at p which he denotes as $v_{p}(f)$ I was not convinced it is true so I tried it out for say $f(x)=(z-p)^n$ and I did get that it was the order of f at p. But I am curious as to why this is true in general.
My last 2 questions is Serre first considers the contour on the boundary of D where there are no zeroes nor poles on the boundary except for $i, \rho, -\rho$. Then he considers the case where there may be additional poles on {$Re(z)=-\frac{1}{2}$ and $Im(z)>\frac{\sqrt{3}}{2}$} and states that we do the same thing but with a change in the contour with a circle around $\lambda$ and $T\lambda$. My first question is if the pole is at $\lambda$, why are we considering $T\lambda$?
Also, why does Serre not consider the possibilities of poles and zeroes on B' to C and C' to D in his contour diagram. He only considers the possibility of additional poles from B to A and E to D'.
For the 1st question, it is the definition of "$f(z)$ is a meromorphic at the cusp " :
by $1$-periodicity and meromorphic on the upper half-plane : $g(q) = f(\frac{\log q}{2i \pi})$ is well-defined (it doesn't depend on the branch of $\log$) and it is meromorphic on $0 < |q| < 1$, and "meromorphic at the cusp" means $g(q) \sim C q^{N}, N \in \mathbb{Z}, C \ne 0$ as $q \to 0$, so it has no pole and zero on $0 < |q| < r$ and it has a Laurent series $$g(q) = \sum_{n=N}^\infty c_n q^n$$ valid on $0 < |q| < r$
Now for $v_p(f)$, it is defined as $$v_p(f) = Res(\frac{f'(z)}{f(z)},p)=\frac{1}{2i\pi}\int_{|z-p| = \epsilon} \frac{f'(z)}{f(z)}dz$$ well-defined whenever $f(z)$ is holomorphic on $0<|z-p|\le\epsilon$ and meromorphic at $z=p$. Write $f(z) = (z-p)^{k}h(z)$ where $h(z)$ is holomorphic and doesn't vanish on $0<|z-p|\le\epsilon$, so that $\frac{f'(z)}{f(z)} = \frac{k}{z-p}+\frac{h'(z)}{h(z)}$ and use that $\int_{|z-p| < \epsilon} \frac{h'(z)}{h(z)}dz=0$ by the Cauchy integral theorem so that $v_p(f) = k$.
For the last questions, everything is explained in Serre's book p.85-86, I can't do better.