Multivalued function

Some of the functions that are useful in practical applications are not uniquely defined. Examples of such functions are nth roots of a complex variable for n 1. For a specific example, consider the case when n = 2  

FIGURE 13.8 Evaluation of fix) dx by contour integration in the complex plane. Only singularities in the upper half plane contribute. 

Thus far we have considered single-valued functions, which are uniquely specified by an independent variable z. The simplest counterexample is the square root fz, which is a two-valued function. Even in the real domain, V4 can equal either 2. When the complex function /z is expressed in polar form 

FIGURE 13.9 Representations of Riemann surface for f z) = The dashed segments of the loops lie on the second Riemann sheet. 

FIGURE 13.10 Schematic representation of several sheets of the Riemann surface needed to cover the domain of a multivalued function such as In z or sin z. 

The Riemann surface for the cube root f(z) = z comprises three Riemann sheets, corresponding to three branches of the function. Analogously, any integer or rational power of z will have a finite number of branches. However, an irrational power such as f(z) = z = will not be periodic in any integer multiple of In and will hence require an infinite number of Riemaim sheets. The same is true of the complex logarithmic function 

it is convenient on multiplication or division to use Euler s formula, since we have shown in Eq. 9.15 that s = I KcosO + isind), hence from Eq. 9.26 

We see it is much easier to add emd subtract angles than direct multiplication of complex numbers. These can be reduced to real and imaginary parts by reversing the process using Eq. 9.26 

A peculiar behavior pattern that arises in complex variables must be recognized very early, and that is the multivalued behavior exhibited by certain functions. We next consider the most elementary such case, the square root function 

In analysis of real variables, no particular problems arise when taking square roots. However, considerable circumspection must be given when s is complex. 

any selected value of r and o will yield a value of the complex variable s = a + id), and this will produce two functions namely, 

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Equation (4) holds generally at the face center but is valid over the whole face if the crystal point group contains a reflection plane through the zone center that is parallel to the face. It also holds for all k vectors that terminate on a line in the BZ face that is parallel to a binary axis. The E(k) may be described either by a singlevalued function of k (with k > 0), which is called the extended zone scheme, or by a multivalued function of k within the first BZ, the reduced zone scheme (see Figure 17.2). 

These points are the branch points of the multivalued function sinh ]z(t) in the complex t plane. The one-kink action equals... 

Since 0 < a < 1 the exponent in Eq. (137) is 1 — a > 0. The mathematical implication is that M(p) (137) is a multivalued function of the complex variable p. In order to represent this function in the time domain, one should select the schlicht domain using supplementary physical reasons [135]. These computational constraints can be avoided by using the Riemann-Liouville fractional differential operator oDlt a [see definitions (97) and (98)]. Thus, one can easily see that the Laplace transform of... 

If the polarization curve is obtained under conditions of constant current (galvanostatlcally) curves such as AFGE or AFDE are usually obcained and it is found that the current-voltage behavior is more time dependent. In many cases periodic fluctuations occur (34,78). Since the potential is a multivalued function of the current, this behavior is readily understandable. 

In the theoretical description of regular polymers, the monoelectronic levels (orbital energies in the molecular description) are represented as a multivalued function of a reciprocal wave number defined in the inverse space dimension. The set of all those branches (energy bands) plotted versus the reciprocal wave number (k-point) in a well defined region of the reciprocal space (first Brillouin zone) is the band structure of the polymers. In the usual terminology, we note the analogy between the occupied levels and the valence bands, the unoccupied levels and the conduction band. 

However, because the logarithm of l/(rj) is a multivalued function, there are, in fact, an infinite number of effective Hamiltonians which yield the correct propagator 1/(tj) = exp - Criteria for the choice of... 

Plots in Fig. 5.16 also indicate that, over certain ranges of /x, u (/x) is a multivalued function where the lowest value of u obviously corresponds to the thermodynamically stable morphology (i.e., phase) the others are only metastable. Metastability ends (i.e., the confined fluid becomes unstable) if the inequality in Eq. (1.82) can no longer be satisfied. The reader should realize that in general metastability in MC simulations is an artifact caused by the limited system size and insufficient length of the Markov chain (i.e., the finite computer time available) [184]. Metastability would not be observed in an infinite system where the evolution of the system could be followed indefinitely. In other words, metastability vanishes in the thermodynamic limit. 

Of course, the approximations made by the spin wave theory are reasonable at very low temperatures only, and thus it is plausible that this line of critical temperatures terminates at a transition point 7 Kt, the Kosterlitz-Thouless (1973) transition, while for T > 7kt one has a correlation function that decays exponentially at large distances. This behavior is recognized when singular spin configurations called vortices (fig. 33 Kawabata and Binder, 1977) are included in the treatment (Berezinskii, 1971, 1972). Because 0(x) is a multivalued function it is possible that a line integral such... 

Monson and Myers have derived a formulation of the thermodynamics of adsorption in terms of excess variables, which yields a similar expression for the excess isosteric heat of adsorption evaluated at constant excess density. The excess differential enthalpy, however, exhibits a singularity close to the maximum of the excess isotherms because the excess density becomes a multivaluate function of pressure. ... 

The process of shifting the domain of a Taylor series is known as analytic continuation. Fig. 13.4 shows the circles of convergence for /(z) expanded about z = 0 and about z = 1. Successive applications of analytic continuation can cover the entire complex plane, exclusive of singular points (with some limitations for multivalued functions). 

By monitoring T using a photomultiplier, R can be determined. Note that the sign of R is unknown since T varies with the sine squared of R hence, one must determine the sign by another method. Also T is a multivalued function of so that the order of the retardation must be established via another route. 

Multivalued Functions 335 Euler formula arises by combining the trigonometric functions... 

Inversion Theory for Multivalued Functions The Second Bromwich Path 379... 

We next consider how to cope with the inversion of transforms containing multivalued functions. To illustrate this, we consider inverting... 

The set of points that reside on a two-dimensional interface in three-dimensional space can, at times, be given explicitly in a certain convenient coordinate system, e.g., z = z x,y) in a Cartesian system or r = r(6,(j)) in spherical coordinates. However, such a description is not always practical. For instance, describing a spherical surface of radius a in a Cartesian system would require a multivalued function z = ... 

However, this field is not a globally Hamiltonian one because the polar angle (x, y) is a multivalued function on a plane without a point, and the above-mentioned local Hamiltonians cannot be "sewed into one smooth single-valued function. This results from the fact that the symplectic manifold R 0 is not simply-connected. 

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