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Transforms: Laplace, Fourier, Z, and Hilbert
Published in A. David Wunsch, ® Companion to Complex Variables, 2018
To establish a branch of a multivalued function, we require not only a branch cut (or cuts) but also the numerical value of the function at one point. Suppose s = 2 or any real value greater than 1. Using s = 2 in Equation 6.1, with our given f(t) = et/t must result in a positive real value, and this completes the specification of the branch of F(s) = π(s−1)1/2, because F(2) = π.
Complex Representations of Functions
Published in Russell L. Herman, A Course in Mathematical Methods for Physicists, 2013
We have seen that some complex functions inherently possess multivaluedness; that is, such “functions” do not evaluate to a single value, but have many values. The key examples were f (z) = z1/n and f (z) = ln z. The nth roots have n distinct values, and logarithms have an infinite number of values as determined by the range of the resulting arguments. We mentioned that the way to handle multivaluedness is to assign different branches to these functions, introduce a branch cut, and glue them together at the branch cuts to form Riemann surfaces. In this way, we can draw continuous paths along the Riemann surfaces as we move from one Riemann sheet to another.
Temporal and spatial frequency domain representation. Interpretation of the temporal transform
Published in Edward J. Rothwell, Michael J. Cloud, Electromagnetics, 2018
Edward J. Rothwell, Michael J. Cloud
Many interesting techniques may be used to compute the inversion integral appearing in (4.399) and in the other expressions obtained in this section. These include direct real-axis integration and closed contour methods using Cauchy’s residue theorem to capture poles of the integrand (which often describe the properties of waves guided by surfaces). Often it is necessary to integrate around branch cuts to satisfy the hypotheses of the residue theorem. When the observation point is far from the source, we may use the method of steepest descents to obtain asymptotic forms for the fields. The interested reader should consult Chew [35], Kong [108], or Sommerfeld [116].
Generalised technique for calculation of plane director profiles in bounded nematic liquid crystals
Published in Liquid Crystals, 2018
O.S. Tarnavskyy, M.F. Ledney, A.I. Lesiuk
Here, since the function in braces is multivalued, the appropriate branch must be chosen. To separate branches, one must make branch cuts between branch points. All these branch cuts, except one related to the defect at the point , should be made outside the region G or along parts of the contour . Corresponding branch cuts in the complex w-plane will lie in the lower half plane . A branch cut between points and should be made. It corresponds to the branch cut between points and in the complex plane w. Then the necessary branch that satisfy boundary conditions is chosen.