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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 Variables
Published in William S. Levine, Control System Fundamentals, 2019
A singular point of a function f(z) is a value of z at which f(z) fails to be analytic. If f(z) is analytic in a region Ω, except at an interior point z = a, the point z = a is called an isolated singularity. For example, f(z)=1z−a.
Complex Analysis
Published in Abul Hasan Siddiqi, Mohamed Al-Lawati, Messaoud Boulbrachene, Modern Engineering Mathematics, 2017
Abul Hasan Siddiqi, Mohamed Al-Lawati, Messaoud Boulbrachene
An isolated singularity z = z0 of f(z) is called a removable singularity if the principal part in the Laurent expansion is zero, that is, all coefficients a−k in (8.35) are zero. An isolated singularity z = z0 is called a pole if the principal part in the Laurent expansion contains a finite number of non-zero terms. It is called a pole of ordern if it has n non-zero terms.
Fundamental solutions for Schrödinger operators with general inverse square potentials
Published in Applicable Analysis, 2018
Huyuan Chen, Suad Alhomedan, Hichem Hajaiej, Peter Markowich
where , the fundamental solution of in with Dirichlet boundary condition and is the Green operator defined by the Green kernel . A super bound is constructed to control this sequence using the hypothesis that in (V). This super bound also provides estimates for the singularity at the origin and the decay at infinity of the minimal fundamental solutions. The isolated singularity (1.11) at origin is also motivated by the classification of isolated singularities in [13, Proposition 4.2].
Some remarks on global analytic planar vector fields possessing an invariant analytic set
Published in Dynamical Systems, 2021
Let and assume without loss of generality that . Let the complex extensions of H be square-free, that is, in its local factorization in a neighborhood in of p into irreducible factors no non-unit factor has a multiplicity larger than one. In other words, the multiplicities for all . Then p is an algebraically isolated singularity of dH. In particular statement (ii) of Theorem 3.3 holds in a neighborhood of p and consequently .
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
Notice that f, g and h have an isolated singularity at z=0. Moreover, from the phase portraits, we can guess that f has a pole of order 1, g has a pole of order 2 and h has a pole of order 3. To confirm this we can calculate the Laurent series representation centred at 0. First observe that Thus we can see that f has a simple pole. On the other hand then g has a pole of order 2. Finally, h has a pole of order 3 since