Doubt in proof of valence formula of entire modular forms

Question

I am self studying analytic number theory from lecture notes of Peter Bruin and Sanders Dahmen.

I have a doubt in proof when the authors writes the identities involving order of zeroes. I am posting the image highlighting the the part I have doubt.

Image -

My doubt is ->How the authors write the last 3 identities in the 2 nd image which are highlighted.

Can someone please help by explaining any 1 identity.

Answer 1

$\int_{C'}^C$ is an integral over a simple closed-loop around the point $SL_2(Z)i$ of the modular curve, thus for $f$ meromorphic and $SL_2(Z)$ invariant, $$\int_{C'}^C\frac{f'}f =2i\pi \ ord_{SL_2(Z)i}(f)$$ (where $ord_i(f) = 2ord_{SL_2(Z)i}(f)$)

$\int_{B'}^B+\int_{D'}^D$ is an integral over a simple closed-loop around $SL_2(Z)e^{2i\pi /3}$.

To extend those things to modular forms $f\in M_k(SL_2(Z))$, we need to let the radius of the piece of circles $\to 0$,

or arg that $g=f/E_k$ is $SL_2(Z)$ invariant and $g'/g= f'/f-E_k'/E_k$.