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