# Complex analysis

## z-Transformation

The infinite impulse response is a property of a filter of digital signal processing. The filter response of the impulse signal does not end infinitely. The filter can be represented in either the time domain or frequency domain. Time-domain filter modifies time-domain input signal to time-domain output signal. The frequency domain lets us understand or to design the effect of the filter. The discrete Fourier transformation $$\Sigma^{N-1}_{n=0} x(n)e^{-i2\pi nm/N}$$ transforms time domain to frequency domain in the finite impulse response.

## 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 $$\Gamma = g(x)$$.