# 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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For instance, the orthogonality of vectors [[bar.e].sup.1] and [[bar.e].sub.3] (and from Cauchy's theorem [18], [[bar.e].sup.3] and [[bar.e].sub.1]) implies that [[bar.e].sup.1] and [[bar.e].sup.3] are conjugate directions.
From Cauchy's theorem, for T [member of] C, we have
Now, remember that the radius a in the Cauchy's theorem is still arbitrarily.
by Cauchy's formula and Theorem 5.1 since we may change the paths of integration to circles by Cauchy's theorem, (6.3) for [u.sub.[+ or -]] and the definition of [A.sub.exp,[+ or -]].
He continues with Abel's theorem, the gamma function, universal covering spaces, Cauchy's theorem for non-holomorphic functions and harmonic conjugates.
the integral with respect to [OMEGA] is now evaluated by closing the contour in the lower half-plane because the integrand has behavior as O([[OMEGA].sup.-2]) for large [OMEGA] and using Cauchy's theorem. This gives

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