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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.
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.
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.
This terminology is also used in complex analysis, as complex number can be desrcibed as vectors in the complex plane.Sources and sinks are associated with zeros and poles of meromorphic function, representing inflows and outflows in a harmonic function. A complex function is defined to a source or a sink if it has a pole of order 1. [14]
The zeros and poles of the Selberg zeta-function, Z(s), can be described in terms of spectral data of the surface. The zeros are at the following points: For every cusp form with eigenvalue s 0 ( 1 − s 0 ) {\displaystyle s_{0}(1-s_{0})} there exists a zero at the point s 0 {\displaystyle s_{0}} .
This time, the numerator is zero at z K = 0, giving K zeros at z = 0. The denominator is equal to zero whenever z K = α. This has K solutions, equally spaced around a circle in the complex plane; these are the poles of the transfer function. This leads to a pole–zero plot like the ones shown below.
The theorem is usually used to simplify the problem of locating zeros, as follows. Given an analytic function, we write it as the sum of two parts, one of which is simpler and grows faster than (thus dominates) the other part. We can then locate the zeros by looking at only the dominating part.
The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles.