### Abstract

The divergent integral , for −∞ < a < x0 < b < ∞ and n = 0, 1, 2, …, is assigned, under certain conditions, the value equal to the simple average of the contour integrals ∫C^{±}f(z)(z − x0)^{−n−1}dz, where C^{+} (C^{−}) is a path that starts from a and ends at b and which passes above (below) the pole at x0. It is shown that this value, which we refer to as the analytic principal value, is equal to the Cauchy principal value for n = 0 and to the Hadamard finite-part of the divergent integral for positive integer n. This implies that, where the conditions apply, the Cauchy principal value and the Hadamard finite-part integral are in fact values of absolutely convergent integrals. Moreover, it leads to the replacement of the boundary values in the Sokhotski-Plemelj-Fox theorem with integrals along some arbitrary paths. The utility of the analytic principal value in the numerical, analytical, and asymptotic evaluations of the principal value and the finite-part integral is discussed and demonstrated.

Received 04 December 2015
Accepted 23 February 2016
Published online 09 March 2016

Acknowledgments:
This work was funded by the UP System Enhanced Creative Work and Research Grant (ECWRG 2015-2-016).

Article outline:

I. INTRODUCTION
II. THE ANALYTIC PRINCIPAL VALUE INTEGRAL
III. THE CAUCHY PRINCIPAL VALUE AND THE FINITE PART INTEGRALS AS VALUES OF THE ANALYTIC PRINCIPAL VALUE
IV. THE SOKHOTSKI-PLEMELJ-FOX THEOREM AND THE ANALYTIC PRINCIPAL VALUE
V. THE CAUCHY PRINCIPAL VALUE AND THE FINITE-PART INTEGRAL INVOLVING FUNCTIONS WITH ENTIRE COMPLEX EXTENSIONS
A. The Cauchy principal value and the finite-part integral
B. The analytic principal value
C. Example
VI. CONCLUSION

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