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In this case a point that is neither a pole nor a zero is viewed as a pole (or zero) of order 0. A meromorphic function may have infinitely many zeros and poles. This is the case for the gamma function (see the image in the infobox), which is meromorphic in the whole complex plane, and has a simple pole at every non-positive integer.
A pole-zero plot shows the location in the complex plane of the poles and zeros of the transfer function of a dynamic system, such as a controller, compensator, sensor, equalizer, filter, or communications channel. By convention, the poles of the system are indicated in the plot by an X while the zeros are indicated by a circle or O.
After applying this rule, the zero poles should be neglected, i.e. if there are no other unstable poles, then the open-loop transfer function () should be considered stable. If the open-loop transfer function G ( s ) {\displaystyle G(s)} is stable, then the closed-loop system is unstable, if and only if, the Nyquist plot encircle the point −1 ...
The simple contour C (black), the zeros of f (blue) and the poles of f (red). Here we have ′ () =. In complex analysis, the argument principle (or Cauchy's argument principle) is a theorem relating the difference between the number of zeros and poles of a meromorphic function to a contour integral of the function's logarithmic derivative.
For any hyperbolic surface of finite area there is an associated Selberg zeta-function; this function is a meromorphic function defined in the complex plane. The zeta function is defined in terms of the closed geodesics of the surface. The zeros and poles of the Selberg zeta-function, Z(s), can be described in terms of spectral data of the surface.
In complex analysis, a branch of mathematics, an isolated singularity is one that has no other singularities close to it. In other words, a complex number z 0 is an isolated singularity of a function f if there exists an open disk D centered at z 0 such that f is holomorphic on D \ {z 0}, that is, on the set obtained from D by taking z 0 out.
The order of vanishing is a generalization of the order of zeros and poles for meromorphic functions in complex analysis. For example, the function ( z − 1 ) 3 ( z − 2 ) ( z − 1 ) ( z − 4 i ) {\displaystyle {\frac {(z-1)^{3}(z-2)}{(z-1)(z-4i)}}} has zeros of order 2 and 1 at 1 , 2 ∈ C {\displaystyle 1,2\in \mathbb {C} } and a pole of ...
Intuitively, the minimum-phase part of a general causal system implements its amplitude response with minimal group delay, while its all-pass part corrects its phase response alone to correspond with the original system function. The analysis in terms of poles and zeros is exact only in the case of transfer functions which can be expressed as ...