Complex analysis

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 ${\Sigma }_{n=0}^{N-1}x\left(n\right)e^{-i2\pi nm/N}$ transforms time domain to frequency domain in the finite impulse response.

Cauchy’s integral formula defines analytic function evaluation with path integral with denominator translation at evaluation point ($\frac{1}{z-a}$). $f\left(a\right)=\frac{1}{2\pi i}{\oint }_{\gamma }\frac{f\left(z\right)}{z-a}dz$ $f^{\left(n\right)}\left(a\right)=\frac{n!}{2\pi i}{\oint }_{\gamma }\frac{f\left(z\right)}{{(z-a)}^{n+1}}dz$ Cauchy’s integral formula is a limit of path. ${lim}_{r\to 0}\gamma :|z-z_0|=r$ Taylor series evaluated a analytic function by approximation at an open disc $D(z_0,r)$. $f\left(x\right)={\sum }_{n=0}^{\infty }{a}_n(x-b{)}^n$ $\frac{f^{\left(n\right)}\left(b\right)}{n!}=a_n$

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