Complex variables singularities

For the sake of simplicity, we will consider an analytic function / = f(z) of a single complex variable z. It is well known that except for the integer functions and the meromorphic functions defined in the entire complex plane, such an analytic function usually has a natural domain restricted by a boundary of singularities, over which it cannot be analytically continued. This means that even in the case of complex scaling v = v(rj)—for certain values of 11—the transformation may take the variable rjz outside the domain of analyticity, and the operation is then meaningless. As an example, we may consider the analytic functions / = f(z) which are defined only within the unit circle z < 1, with this circle as natural boundary, and it is then evident that the complex scaling is meaningful only for 7 < 1. 

An important property of a power series is its radius of convergence, determined by the singularity closest to the origin. This, of course, is not known in the present cases but we may attempt to estimate it as follows (Fernandez [17]). The perturbation expansion (37) is a real power series in X, so we choose the singular points of ni (X) to be the complex conjugate pair Xo and Xq (now allowing X to be a complex variable) and consider the function... 

We will assume that in the upper half-plane of complex variable m there are no singularities except the branch points m = k a and m = k2a. 

These are the Cauchy-Riemann conditions, and when they are satisfied, the derivative dw/ds becomes a unique single-valued function, which can be used in the solution of applied mathematical problems. Thus, the continuity property of a complex variable derivative has two parts, rather than the one customary in real variables. Analytic behavior at a point is called regular, to distinguish from nonanalytic behavior, which is called singular . Thus, points wherein analyticity breaks down are referred to as singularities. Singularities are not necessarily bad, and in fact their occurrence will be exploited in order to effect a positive outcome (e.g., the inversion of the Laplace transform ). 

Figure 4. Complex plane of the variable s. The vertical axis Rei is the axis of the rates or complex frequencies. The horizontal axis Imr is the axis of real frequencies to. The resonances are the poles in the lower half-plane contributing to the forward semigroup. The antiresonances are the poles in the upper half-plane contributing to the backward semigroup. The resonances are mapped onto the antiresonances by time reversal. Complex singularities such as branch cuts are also possible but not depicted here. The spectrum contributing to the unitary group of time evolution is found on the axis Re = 0.

We will make use of Cauchy s theorem, according to which the integral value of an analytical function does not change under deformation of an integration contour if it does not intersect singularities on the complex plane of variable to. It is clear that deforming the contour of integration in the upper half-plane (Im m > 0) exponent e with an increase of Im TO tends to zero. 

It may be used in the integral Eq. (3.14) instead of the form Eq. (3.17), provided the integration of the energy variable is performed as a contour integral in the complex E-plane. An appropriate contour is chosen such that it bypasses the singularities at E = e on the real axis. Figure 3.1 displays an acceptable contour for the case that fk > fi when ck < cj. 

This analysis, based on the single variable T, shows that multiple critical transitions ("ignition and extinction) and regions of bistability are possible in non-isothermal reaction when complex Kinetics occur. It cannot reveal the types of behaviour associated with the singularities of the Gray and Yang scheme, but the proper two-dimensional (T-[x]) stability analysis establishes the following ... 

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