Sokhatsky-Weierstrass theorem: Facts, Discussion Forum, and Encyclopedia Article

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Sokhatsky-Weierstrass theorem

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	Topics Sokhatsky-Weierstrass theorem Topic Home The Sokhatsky-Weierstrass theorem (also spelled Sokhotsky-Weierstrass theorem, and also called the Weierstrass theorem, although the latter term has several, more common, alternate meanings) is a theoremTheorem A theorem is a proposition that has been or is to be proved on the basis of explicit assumptions.... in complex analysisComplex analysis Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use i... , which helps in evaluating certain Cauchy-type integrals, among many other applications. It is often used in physics, although rarely referred to by name. The theorem is named after Yulian SokhotskiYulian Vasilievich Sokhotski Yulian Karl Vasilievich Sokhotsky was a Russian mathematician.... and Karl WeierstrassKarl Weierstrass Karl Theodor Wilhelm Weierstrass was a German mathematician who is often cited as the "father of modern analysis".... . Proof of the theorem A simple proof is as follows. For the first term, we note that is an approximate identityApproximate identity In functional analysis, a right approximate identity in a Banach algebra A is a net... , and therefore approaches a Dirac delta functionDirac delta function The Dirac delta or Dirac's delta, often referred to as the unit impulse function and introduced by the British theoret... in the limit. Therefore, the first term equals . For the second term, we note that the factor approaches 1 for \|x\| >> e, approaches 0 for \|x\| << e, and is exactly symmetric about 0. Therefore, in the limit, it turns the integral into a Cauchy principal valueCauchy principal value In mathematics, the Cauchy principal value of certain improper integrals is defined as either... integral. Physics application In quantum mechanicsQuantum mechanics Quantum mechanics is a first quantized quantum theory that supersedes classical mechanics at the atomic and subatomic levels... and quantum field theoryQuantum field theory Quantum field theory is the quantum theory of fields.... , one often has to evaluate integrals of the form where E is some energy and t is time. This expression, as written, is undefined (since the time integral does not converge), so it is typically modified by adding a negative real coefficient to t in the exponential, and then taking that to zero, i.e.: where the latter step uses this theorem. uses this theorem.