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Philosophical Transformation Essential to Reverse Engineer Consciousness
Published in Anirban Bandyopadhyay, Nanobrain, 2020
What is a singularity? A singularity is a gap in the phase space, where the phase structure of a typical biomaterial is undefined: the output is irrelevant to the input. In these conditions, a system resonantly vibrates, emits, or absorbs the signal of a particular frequency. There are many different types of singularities; for example, phase singularity, polarization singularity, and amplitude singularity. Singularity points are the corners of geometric shapes in the phase structure of a spiral or vortex. A system point passes through the corners one by one; if in a loop, it’s a clock. The signal bursts when the system point passes through the singularity points located on the loop that defines a clock. The singularity that Feynman eschewed in his renormalization (Feynman, 1949; Cao and Schweber, 1993), the 3D phase structure, holds the Feynman paths as a minor subset in the geometric shapes. Thus, the universe in the time crystal model blinks, spontaneously creating a vertical Turing tape in the cell of a horizontal Turing tape (see Figure 1.5a).
Analysis
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
Given a point z0 where f(z) is either analytic or has an isolated singularity, the residue of f(z) is the coefficient of (z - z0)-1 in the Laurent series expansion of f(z) at z0, or Res(z0)=b1=12πi∫Cf(z)dz. $$ {\text{~Res~}}(z_{0} ) = b_{1} = \frac{1}{{2\pi i}}\mathop \smallint \limits_{C}^{{}} f(z)dz. $$
Complex Representations of Functions
Published in Russell L. Herman, A Course in Mathematical Methods for Physicists, 2013
Another way to find the residue of a function f (z) at a singularity z0 is to look at the Laurent series expansion about the singularity. This is because the residue of f (z) at z0 is the coefficient of the (z − z0)−1 term, or c−1 = b1. The residue of f (z) at z0 is the coefficient of the (z − z0)−1 term, c−1 = b1, of the Laurent series expansion about z0.
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
Figure 9 shows the enhanced phase portrait of the function defined in the square and . We notice that f has a singularity at z=0 but in this case the plot does not show isochromatic lines meeting at that point. This indicates that the singularity might be removable.