Cauchy’s integral

This last representation is completely equivalent to the analytidty of t(ai) in Im 0 and the statement that a,t(a>) go to zero as u - oo. The analyticity property in turn is a direct consequence of the retarded or causal character of T(t), namely that it vanishes for t > 0. If t(ai) is analytic in the upper half plane, but instead of having the requisite asymptotic properties to allow the neglect of the contribution from the semicircle at infinity, behaves like a constant as o> — oo, we can apply Cauchy s integral to t(a,)j(o, — w0) where a>0 is some fixed point in the upper half plane within the contour. The result in this case, valid if t( - oo is... 

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... 

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... 

Figure 22.1 Domain of integration for application of Cauchy s integral formula. Poles are placed at frequencies on the real frequency axis.

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,... 

Example A.2 Special Case of Cauchy s Integral Formula Find the numerical value for the integral f z — a) dzfor the case where z = ais inside the domain. 

This result is a special case cif Cauchy s Integral Formula. 

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,... 

Remember A.3 The derivation of the Kramers-Kronig relations in Section 22.1 makes use of Cauchy s Integral Formula far evaluation cf a function at a singularity, given as Example A.2. 

Cauchy s integral formula can be differentiated with respect to zo any number of times to give... 

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... 

Cauchy s Integral Formula states that if F z) is analytic within and on a closed contour C, then... 

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