Note: In mathematics, there is no field called "Residue Theory". There is only a "Calculus of Residues", belonging to the field of "Complex Analysis".

Residue Theory is fairly short, with only a few methods; and is basically a glorified way of saying, "Finding the coefficients when expanding fractions." Although this small subject blossoms into other areas of mathematics, particularly in simplifying integrals, and in analyzing singularities in limits, the real-number system, or the complex number system.

Knowing the following ideas is more-or-less required:

Power/Taylor/Laurent Series

Limits/Derivatives/Integrals

And for the more advanced techniques in sections 3 and 4: