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Complex Representations of Functions
Published in Russell L. Herman, A Course in Mathematical Methods for Physicists, 2013
There are three types of singularities: removable, poles, and essential singularities. They are defined as follows: If f (z) is bounded near z0, then z0 is a removable singularity.If there are a finite number of terms in the principal part of the Laurent series of f (z) about z = z0, then z0 is called a pole.If there are an infinite number of terms in the principal part of the Laurent series of f (z) about z = z0, then z0 is called an essential singularity.
Analysis
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
A singular point or singularity of the function f(z) is any point at which f(z) is not analytic. An isolated singularity of f(z) at z0 may be classified as one of:A removable singularity if and only if all coefficients bn in the Laurent series expansion of f(z) at z0 vanish. This implies that f(z) can be analytically extended to z0.A pole is of order m if and only if (z - z0)mf(z) , but not (z - z0)m-1f(z) , is analytic at z0, (i.e., if and only if bm ≠ 0 and 0 = bm+1 = bm+2 = … in the Laurent series expansion of f(z) at z0). Equivalently, f(z) has a pole of order m if 1/f(z) is analytic at z0 and has a zero of order m there.An isolated essential singularity if and only if the Laurent series expansion of f(z) at z0 has an infinite number of terms involving negative powers of z - z0.
Complex analysis
Published in John P. D’Angelo, Linear and Complex Analysis for Applications, 2017
The definition of essential singularity at p is equivalent to: the Laurent series of f has cn ≠ 0 for infinitely many negative n. For example, e−1z=1−1z+12z2−16z3+⋯.
The use of phase portraits to visualize and investigate isolated singular points of complex functions
Published in International Journal of Mathematical Education in Science and Technology, 2019
The advantage of visualizing essential singularities is not only that students can observe the behaviour of the function near this kind of singularity, but also it might help them to explore and comprehend more abstract mathematical results such as the Great Picard Theorem, which tells us that any analytic function with an essential singularity at takes on all possible complex values (with at most a single exception) infinitely often in any neighbourhood of (Krantz, 2004, pp. 28–30).