Cauchy's Integral Formula:
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Similarly, the following theorem shows that the value of the $n$-th derivative $f^{n}(z)$ can be represented by a certain contour integral involving the values of $f(z)$ on $C$.

Cauchy's Integral Formula for $n$-th Derivative: Let $f$ be analytic in the simply connected domain $D$, and let $C$ be a simple positively oriented contour that lies in $D$. Let $f^{n}(z)$ denote the $n$-th derivative of $f(z)$. If $z_{0}$ is a point that lies interior to $C$, then MATH

Cauchy's Estimate: Let $f(z)$ be analytic on and inside the circle MATH. Let MATH. Then, MATH

Theorem: If a function $f$ is analytic at a point, then its derivative of all orders are also analytic functions at that point.

Corollary: If a function MATH is analytic at a point $z = x+iy$, then the component functions MATH and MATH have continuous partial derivatives of all orders at that point.

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