Cauchy’s integral formula

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... 

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... 

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

Another way to introduce fractional operators is by generalizing Cauchy s formula for a w-fold integration over a fixed time interval (a,t) ... 

Hereinafter some few preliminary definitions of fractional calculus are introduced for clarity sake s as well as for introducing appropriate symbologies. Let us start with the definition of the Cauchy formula for multiple integrals... 

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