Cauchy’s integral theorem

As shown in equation (22.10), the real part of the impedance tends toward a finite value as frequency tends toward infinity. The transfer function Z x) — Zr,oo tends toward zero with increasing frequency. As Z(x) is analytic, Cauchy s integral theorem, given in Appendix A as Theorem (A.2), can be written as... 

Sometimes we would like to resolve system stability without modeling impedances or determining the zeros and poles of the impedance. This can be done using the Nyquist stability criterion [587, 588, 596] developed from the theory of complex functions and Cauchy s integral theorem, which can be stated as follows an electrochemical system is stable if and only if the number of clockwise encirclements ( N) of the origin of the Z (—Z") plane, going from low to high frequencies, equals the... 

A function G that satisfies equation (22.29) can be shown, by use of Cauchy s Integral Formula (Theorem A.3), to be a causal transform. The properties of G implicit in Theorems 22.1-22.3 and equation (22.29) allow derivation of dispersion relations... 

If the radii ei and 2 of the semicircular paths 71 and 72 approach zero, the term 1/(x - - to) dominates along path 71, and l/(x — u ) is the dominant term along path 72. From an application of Cauchy s Integral Formula, Theorem (A.3), to a half-circle,... 

Theorem A.3 (Cauchy s Integral Formula) If f z) is analytic in a simply connected domain D, and ifC is a simple positively oriented (counterclockwise) closed contour that lies in D, then, for any point zq that lies interior to C,... 

Example A.l Application of Cauchy s Theorem Example A.2 Special Case of Cauchy s Integral Formula Example A.3 Poles on a Real Axis... 

Now, if n is a positive integer, or zero, the above is obviously in accordance with Cauchy s First theorem, since then fis) = s" is analytic for all finite values of s. However, if n becomes a negative integer, the s" is clearly not analytic at the point s = 0. Nonetheless, the previous result indicates the closed integral vanishes even in this case, provided only that n — 1. Thus, only the simple pole at the origin produces a finite result, when the origin is enclosed by a closed contour. 

According to Cauchy s theorem, plus the integral over the large half-circle Cm... 

This integral may be evaluated by means of Cauchy s theorem.1) Its value is found to be... 

If At 6 E is a function of a real or complex parameter t, we can define differentiation and integration with respect to t, usual rules of operations being applicable to them. Also regularity (analyticity) of At can be defined and Cauchy s, Taylor s and Laurent s theorems are extended to these regular functions. 

In order to derive the adjoint operator u we will now study the expression for a function f(x) in the domain D u) and another function g x) in L2. Putting z = r] x and using Cauchy s theorem about contour integrals, one obtains—provided that the integrand becomes sufficiently small on the outside arcs—that... 

Then one can invoke Jordan s lemma and Cauchy s theorem (see Whittaker Watson (1946)) for the line integral in (2.6.11) that can be converted to the contour integral, as shown in Figure 2.18, with only a single pole indicated at the point Pi. Let us also say that the disturbance corresponding to this pole has a positive group velocity i.e. the associated disturbance propagates in downstream direction. 

The integral around the closed contour is also designated f f z)dz. A major consequence of Cauchy s theorem is that the value of the integral from one point to another is independent of the path. Two paths Ci eind C2 between points A and B are shown in Figure A.5. The contour directions are the same thus. 

Once the Laplace transform u(x,s) of the temperature () (x, /,) which fits the initial and boundary conditions has been found, the back-transformation or so-called inverse transformation must be carried out. The easiest method for this is to use a table of correspondences, for example Table 2.3, from which the desired temperature distribution can be simply read off. However frequently u(x,s) is not present in such a table. In these cases the Laplace transformation theory gives an inversion theorem which can be used to find the required solution. The temperature distribution appears as a complex integral which can be evaluated using Cauchy s theorem. The required temperature distribution is yielded as an infinite series of exponential functions fading with time. We will not deal with the application of the inversion theorem, and so limit ourselves to cases where the inverse transformation is possible using the correspondence tables. Applications of... 

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. 

Now we will derive expressions for the vertical component of the field on the borehole axis when there is an invasion zone and measurements are performed at the far zone. Taking into account that the integrand in eq. 4.136 is an even function we will consider integration along whole axis m and, applying Cauchy s theorem, the contour of integration then will be deformed in the upper part of the complex plane of m without intersection of singularities on this plane. 

According to Cauchy s theorem, 2/ (k) plus the integral over the large half-circle Cm plus the integral over the small half-circle Cj (see Figures la and lb) is equal to the residue. Hence we have to calculate these two integrals on the one hand, and the residue on the other hand. 

Inserting Eq. (117) into Eq. (114), deforming the contour in the complex -plane, and using Cauchy s theorem to evaluate the above integral leads finally to... 

The most important applications of Cauchy s theorem involve functions with singular points. Consider the integral... 

Taylor s theorem can be derived from the Cauchy integral theorem. Let us first rewrite Eq. (13.31) as... 

To achieve momentum balance in a local form, we have the analogous difficulties with surface integral in the right-hand side of (4.50) as in Sect. 3.3. We therefore use analogical Cauchy s postulate and theorem but concerning here partial tractions and stresses (motivation and deductions are quite analogical as in Sect. 3.3) The Cauchy postulate for partial tractions is... 

Integration Cauchy s Theorem 341 order pole. This simply means that fis) must have contained a term... 

Thus, if any contour is drawn so that the branch point is encircled, then multivalued behavior arises. The principle to ensure analyticity is simple branch points cannot be encircled. There is considerable range and scope for choosing branch cuts to ensure analytic behavior. Now, if no other singularities exist in the contour selected, then Cauchy s First Integral theorem is valid, and we denote the new contour C2 as the second Bromwich path, Br2 hence. 

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