Integration Cauchys Theorem

The order of the pole is thus determined by the minimum value of n necessary to remove it (cancel it). 

For an essential singularity, it is not possible to remove the infinite discontinuity. To see this for the classic example mentioned, expand expCl/ ) in series. 

If we tried to remove the singularity at the origin, by multiplying first by s, then s, then s, and so forth, we would still obtain an infinite spike at s = 0. Since the singularity cannot be removed, it is called an essential singularity. 

To implement this, we shall need to know something about integration in the complex domain. 

Integration in the complex domain is necessarily two-dimensional, since variations in real and imaginary variables occur together. Thus, if we wish to find the integral of some arbitrary complex function /(s), then we must stipulate the values of s such as those traced out by the curve C in Fig. 9.2. 

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According to the Cauchy theorem, the integral is zero if Em Eo, because the singularity Em is not inside F, and it is zero when Em = Eo because the singu "Tity of the denominator is compensated by the numerator. Therefore, R o and K are zero. 

The last integral can be calculated with the use of the Cauchy theorem about integral values. It results in... 

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,... 

The Cauchy theorem may be proved by application of (3.70) to an infinitesimal tetrahedron at considered place x and instant t, the walls of which are formed by coordinate planes and a tangent plane perpendicular to considered n. The estimate of the surface and volume integrals in (3.70) gives (using (3.68))... 

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

When pole singularities exist, along with branch points, the Cauchy theorem is modified to account for the (possibly infinite) finite poles that exist within the Bromwich Contour hence, E res (F(j)e ) must be added to the line integrals... 

This is the Gaussian law of electrostatics in integral and differential form. The latter may be reorganized to express the electric flux density with the aid of the charge density of the dedicated area, as given by Eq. (3.29b), leading to the equivalent of the Cauchy theorem of mechanics from Eq. (3.13) ... 

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. 

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... 

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. 

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,... 

Theorem A.5 (Evaluation of the Cauchy Principal Value of an Integral) If a function f z) is analytic in a simply connected main D, except at a finite number of singular points zi,..., 2jt, iff x) = P x)/Q x) where P(x) and Q(x) are polynomials, Q x) has no zeros, and the degree ofP x) is at least two less than the degree of Q(x), and ifC is a simple positively oriented (counterclockwise) closed contour that lies in D, then... 

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... 

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. 

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