# Cauchy Theorem

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## Cauchy Theorem

a theorem concerned with the expansion of an analytic function into a power series. Suppose f(z) is a function that is single-valued and analytic in a region G, Z0 is an arbitrary (finite) point of G, and ρ is the distance from z0 to the boundary of this region. Then there exists a power series in z – z0 that converges to the function in the interior of the circle ǀz – z0ǀ = ρ: If the boundary of G reduces to the point at infinity, then ρ is infinite. This theorem was established by A. Cauchy (1831), who based it on his representation of an analytic function in the form of the so-called Cauchy integral.

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The following result is a version of Cauchy Theorem in nonsmooth framework.
Among the useful tools from the general theory on slice regular functions, we recall the Cauchy theorem and the formula to compute the derivatives, see [3, Theorems 4.
By Cauchy theorem, we obtain, as in , that its Fourier transform in t is given by
Hence by Cauchy theorem, as in , its Fourier transform in t is given by (2.
iii) The proof is based on the Cauchy theorem with [OMEGA] the closed unit ball B(1) and hence [partial derivative][OMEGA] the unit sphere [S.
k] is left monogenic in B(1), the Cauchy theorem yields

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