What's a good way to find how fast the integral of a function is growing near a pole of the function? Here is what I mean on an example.

Look at 1/z. If I want to find out how fast ∫₀^a 1/(z-ε)dz is growing when ε->0, ε∈C, I can do this:

∫₀^a 1/(z-ε)dz = ln|ε/(a-ε)|=ln|ε|+ln|a|+ε/a+O(ε).

What if I have ∫₀^a f(z)/(z-ε) dz , where f(z) is finite?

What's a good way to find how fast the integral of a function is growing near a pole of the function? Here is what I mean on an example.

Look at 1/z. If I want to find out how fast ∫₀^a 1/(z-ε)dz is growing when ε->0, ε∈C, I can do this:

∫₀^a 1/(z-ε)dz = ln((a-ε)/ε)=-ln(-ε)+ln(a)+ε/a+O(ε).

What if I have ∫₀^a f(z)/(z-ε) dz , where f(z) is finite?

This is what I have in mind. Say ε is a complex number approaching 0, you have a real valued function f(z) defined in [0,1], differentiable and for simplicity assume f(z) is near 1 for all z ∈ [0,1]. What's a good way to calculate the asymptotics of ∫₀¹ f(z)/(z-ε) dz ?

What's a good way to find how fast the integral of a function is growing near a pole of the function? Here is what I mean on an example.

Look at 1/z. If I want to find out how fast ∫₀^a 1/(z-ε)dz is growing when ε->0, ε∈C, I can do this:

∫₀^a 1/(z-ε)dz = ln|ε/(a-ε)|=ln|ε|+ln|a|+ε/a+O(ε).

What if I have ∫₀^a f(z)/(z-ε) dz , where f(z) is finite?