What is the name of this theorem? (complex analysis)

In summary, The problem is to evaluate integrals with simple poles offset by ε above/below the real axis. The answer is the sum of two integrals: the principal value of the integral with ε=0 plus the integral of iπδ(x-x0). The Wikipedia calls it Sokhatsky-Weierstrass theorem and provides a derivation for it. The residue is off by a factor of a half, with a factor of 2 in front of the delta.
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gluons
15
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I am working on a problem to evaluate integrals with simple poles offset by ε above/below the real axis. So something like this

∫ [ f(x) / (x-x0-iε) ]

The answer is the sum of two integrals: the principal value of the integral with ε=0 plus the integral of iπδ(x-x0).

I have done the proof for the answer but my residue is off by a factor of a half (I have a factor of 2 in front of the delta, and I'm not sure why).

Does anyone know the name of this theorem and a place for the derivation?
 
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FAQ: What is the name of this theorem? (complex analysis)

What is the name of this theorem?

The name of this theorem is the Cauchy's Integral Theorem.

What is the significance of the Cauchy's Integral Theorem?

The Cauchy's Integral Theorem is a fundamental result in complex analysis that relates the values of a function within a closed contour to its values on the contour itself. This theorem is important in many areas of mathematics, including the study of complex functions, integration, and contour integration.

Who discovered the Cauchy's Integral Theorem?

The Cauchy's Integral Theorem was discovered by French mathematician Augustin-Louis Cauchy in the early 19th century.

What are the applications of the Cauchy's Integral Theorem?

The Cauchy's Integral Theorem has numerous applications in mathematics and physics, including the evaluation of complex integrals, calculating residues of complex functions, and solving differential equations.

What are the assumptions for the Cauchy's Integral Theorem to hold?

The Cauchy's Integral Theorem holds under the assumption that the function is analytic within and on the boundary of the contour, and that the contour is simple and closed.

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