Cauchy Integral theorem

The Cauchy Integral theorem states that for a function $ f(z) $ which is analytic inside and on a simple closed curve $ C $ in some region $ {\mathcal {R}} $ of the complex Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): z plane, for a complex number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): a inside Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f(a) = \frac{1}{2 \pi i } \int_{C} \frac{f(z)}{z - a} ; dz.

Proof:

For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f(a) a constant, we may rewrite Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): dz = d(z-a) , and noting that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): z - a = | z - a | e^{i \phi} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): d(z-a) = i |z -a | e^{i\phi} \; d\phi , and taking the contour of integration to be a circle of unit radius Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C : |z -a |= 1 , we may write

$ {\frac {1}{2\pi i}}\int _{C}{\frac {f(a)}{z-a}}\;dz={\frac {f(a)}{2\pi i}}\int _{0}^{2\pi }{\frac {i|z-a|e^{i\phi }\;d\phi }{|z-a|e^{i\phi }}}=f(a) $

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By Cauchy's Theorem, we may deform the contour into any closed curve that contains the point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): z - a and the result holds.

For the case when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f(z) is not constant we may write

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \frac{1}{2 \pi i } \int_{C} \frac{f(z)}{z - a} \; dz = \frac{1}{2 \pi i } \int_{C} \frac{f(a)}{z - a} \; dz + \frac{1}{2 \pi i } \int_{C} \frac{f(z) - f(a) }{z - a} \; dz = f(a) + \frac{1}{2 \pi i } \int_{C} \frac{f(z) - f(a) }{z - a} \; dz

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We must show that the second term on the left is identically zero.

In a vanishingly small neighborhood Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): C:|z -a | = \varepsilon

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \frac{1}{2 \pi i } \int_{C} \frac{f(z) - f(a) }{z - a} \; dz \rightarrow \frac{1}{2 \pi i } \int_{C} f^\prime (z) \; dz = 0

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Because the derivative of an analytic function is also analytic, the integral vanishes identically within a neighborhood of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): z = a . By Cauchy's theorem, the contour of integration may be expanded to any closed curve within Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\mathcal R} that contains the point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): z = a thus showing that the integral is identically zero.

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