Complex analysis

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 \(D(z_{0}, r)\). \(f(x) = \sum_{n=0}^\infty a_n(x-b)^n\) \(\frac{f^{(n)}(b)}{n!} = a_n\)

Primitive function

The presence of primitive function is a strong condition that makes a function is analytic in a disc \(D(a,R)\). The meaning is the presence of primitive function is confusing at first to me. If a function is integrable, then integration value and a primitive function can be determined. But in complex analysis this is not the case. In real analysis, the integral interval \([a, b]\) is unique, but in complex analysis the integral interval should be determined by line path \(\Gamma = g(x)\).