Cauchy integral

The function f(t), referred to as the density function, is assumed to be bounded everywhere, except possibly at a finite number of points / = 1,2,. ... In the present context, these points can always be taken to be end points of the arcs included in L, At the endpoints, it may have integrable singular points with 

Furthermore, it is assumed that f t) is Hdlder continuous at each point of L where it is not singular. This very important property is defined as follows for any two points i/i, 1/2. let there exist positive, real constants A, fi such that 

It is easy to show that if // 1, the derivative of/(w) is zero, so that it is a constant. This case is not of great interest, so it is always assumed that /i 1. For // = 1, the Hdlder condition is termed the Lipschitz condition and is obeyed by any differentiable function, and others not in this class. For // 1, the condition implies continuity in the ordinary sense. The case // = 0, which is excluded, is consistent with discontinuity. A function obeying this condition at a point, or on a line, will be described as obeying the H(ji) condition on that set, if n is specified or otherwise just the H condition. 

The Holder condition (A2.2.3) implies that the first term approaches a well-defined integral. 

S and S , respectively. In each of these cases, we deform the contc ur into a small semi-circle around u and consider the limit as this semi-circle shrinks to zero. It is easy to show that 

---

Cauchy-integral method, 219-220 cyclic wave functions, 224-228 modulus and phase, 214-215 modulus-phase relations, 217-218 near-adiabatic limit, 220-224 reciprocal relations, 215-217, 232-233 wave packets, 228-232 multidegenerate nonlinear coupling,... 

Integrals of the type (80) are known as Cauchy integrals and they have well defined properties.25... 

As we have discussed, C (z) is a Cauchy integral and, as a consequence, we immediately have ... 

The passage from the Stieltjes to the Cauchy integral via the Dirac-Lebesgue measure for discretization of inner products 183... 

A simple curve is a closed curve that does not intersect itself. Equation (2.235) is called the Cauchy integral formula. As a consequence of the Cauchy integral formula, we can write... 

To do this it is first convenient to rewrite (62) in the form of a Cauchy integral... 

Loss of Phase in a Quantum Measurement The Cauchy-Integral Method for the Amplitudes Simplified Cases... 

---