Cauchy’s theorem

Stability in the frequency domain 6.4.1 Conformal mapping and Cauchy s theorem... 

As. vi in Figure 6.15(a) is swept clockwise around the contour, it encircles two zeros and one pole. From Cauchy s theorem given in equation (6.46), the number of clockwise encirclements of the origin in Figure 6.15(b) is... 

A frequency domain stability criterion developed by Nyquist (1932) is based upon Cauchy s theorem. If the function F(s) is in fact the characteristic equation of a closed-loop control system, then... 

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... 

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. 

Where, C = C - -G. For an anal3dic function f z) in a domain bounded by a closed contour C, Cauchy s theorem states that,... 

For non-re-entrant particles, according to Cauchy s theorem, d = E dp) and has a maximum value of 1.0 for circles, rectangles and other convex shapes it is therefore very useful for indicating the extent of concavities, y/p, y/ p, y/j p, y/p, y/p and Z where found to show mainly the slimnness of the particles, the best indicators being y/p and Z in that order. y/ p and y/ p were found to correspond poorly with particle morphology. 

Theorem A.2 (Cauchy s Theorem) Iff z) is analytic in a simply connected domain D, and ifC is a simple closed contour that lies in D, then... 

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. 

Remember A.2 The derivation of the Kramers-Kronig relations in Section 22.1 makes use of Cauchy s Theorem, given as Theorem A.2. 

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... 

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... 

For a statement of Cauchy s theorem see Lyusternik, L.A., Convex Figures and Polyhedra, translated from the Russian by T. Jefferson Smith, Dover Publications, New York, 1963, p. 60 ff... 

The value of 6j, the real part of the dielectric constant, may be obtained from the Kramers-Kronig transformation. The basis of this approach is that it can be shown that e(cu) - 1 = (co) - 1 + ie2(co) is analytic in the upper half of the complex plane from Cauchy s theorem, we have, therefore... 

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. 

One may also derive, using also Cauchy s theorem, an expansion of the outgoing Green s function in the energy or k planes in terms of resonant states [17,55]. Such an expansion may be written as... 

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... 

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