The Residue Theorem is a fundamental result in complex analysis. It states that the contour integral of a function f of z around a simple closed contour C equals 2 pi i times the sum of residues at isolated singularities inside the contour.
Let's work through a specific example. We want to evaluate the contour integral of 1 over z squared plus 1, where C is the circle with radius 2 centered at the origin. First, we find the singularities by setting z squared plus 1 equal to zero, giving us z equals plus or minus i. Both singularities are inside our contour since their absolute values are 1, which is less than 2.
Now we calculate the residues at each singularity. For a simple pole at z naught, the residue is the limit as z approaches z naught of z minus z naught times f of z. At z equals i, we get the residue as 1 over 2i. At z equals negative i, we get the residue as 1 over negative 2i, which equals negative 1 over 2i.
Now we apply the Residue Theorem. First, we sum the residues: 1 over 2i plus negative 1 over 2i equals zero. According to the theorem, the contour integral equals 2 pi i times the sum of residues, which is 2 pi i times zero, giving us a final answer of zero. This result occurs because the two residues have equal magnitudes but opposite signs, causing them to cancel out.
To summarize what we've learned: The Residue Theorem provides a powerful method for evaluating contour integrals by focusing on singularities inside the contour. We identify singularities, calculate their residues, sum them up, and multiply by 2 pi i. This technique transforms complex integration into algebraic calculations, making it an essential tool in complex analysis.