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

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

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