# 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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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 4.5.3, 4.5.4] and the definition of axially symmetric s-domain, see [3, Definitions 4.1.4,4.3.1].
The following result is a version of Cauchy Theorem in nonsmooth framework.
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 (2.9).
(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.sup.m-1].
Moreover, as [bar.[M.sub.t]] is right monogenic in B(1), while [M.sub.k] is left monogenic in B(1), the Cauchy theorem yields

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