Cauchy's integral formula and Taylor series

Cauchy’s integral formula defines analytic function evaluation with path integral with denominator translation at evaluation point ($$\frac{1}{z-a}$$).

$$f(a) = \frac{1}{2\pi i} \oint_\gamma \frac{f(z)}{z-a}\, dz\,$$

$$f^{(n)}(a) = \frac{n!}{2\pi i} \oint_\gamma \frac{f(z)}{\left(z-a\right)^{n+1}}\, dz$$

Cauchy’s integral formula is a limit of path.

$$\lim_{r \rightarrow 0}\gamma : \lvert z-z_{0} \rvert = r$$

Taylor series evaluated a analytic function by approximation at an open disc $$D(z_{0}, r)$$.

$$f(x) = \sum_{n=0}^\infty a_n(x-b)^n$$

$$\frac{f^{(n)}(b)}{n!} = a_n$$

Cauchy’s integral and Taylor series evaluate a function at a point $$z$$ in an open disc centered at $$z_{0}$$.

$$z \in D(z_{0}, r)$$

Jun Kang
Clinical Assistant Professor of Hospital Pathology

My research interests include pathology, oncology and statistics.