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Contour Integrals and Residues
You might want to start learning about contour integrals and residues by learning some elementary complex analysis. The contour integrals and residues are (in physics at least) mostly used as a trick to calculate (real!) integrals that would be impossible to calculate otherwise. Here is an example: en.wikipedia.org/wiki/Residue_theorem#Example .
Intuitive explanation to why superconductivity breaks at high temperatures
Actually if I think about it, my question could be rephrased as: why is there a temperature dependence in the gap? Yes, I can see from the gap equation that there is, but how would you interpret that information?