~54~ International Journal of Statistics and Applied Mathematics 2021; 6(5): 54-58 ISSN: 2456-1452 Maths 2021; 6(5): 54-58 © 2021 Stats & Maths www.mathsjournal.com Received: 19-07-2021 Accepted: 21-08-2021 Arup Kumar Saha Department of Education in Science and Mathematics Regional Institute of Education (NCERT), Bhubaneswar, Odisha, India Manoj Kumar Hota Department of Mathematics, Nayagarh Autonomous College, Nayagarh, Odisha, India Prasanta Kumar Mohanty Department of Mathematics, School of Applied Sciences KIIT Deemed to be University, Bhubaneswar, Odisha, India Corresponding Author: Arup Kumar Saha Department of Education in Science and Mathematics Regional Institute of Education (NCERT), Bhubaneswar, Odisha, India Approximate evaluation of complex hyper singular integrals Arup Kumar Saha, Manoj Kumar Hota and Prasanta Kumar Mohanty DOI: https://doi.org/10.22271/maths.2021.v6.i5a.726 Abstract In this paper we develop a method for approximate evaluation of complex hyper singular integrals in the complex plane. The schemes are numerically validated using a set of conventional test integrals. A number of examples is provided to illustrate the the efficiency of the method develop here. Keywords: analytic function, cauchy principal value, Hardamad finite part integral, Teylor’s coefficients 1. Introduction Integrals of the type (,  0 )=∫  () (− 0 )  ;  ∈ ℕ − {1}; (1) are frequently appeared in contour integration, where () is infinitely differentiable function in Ω = { ∈ ℂ: | −  0 | <  = |ℎ|, > 1}; of the complex plane ℂ and  joining the points  0 −ℎ to  0 +ℎ lying in the disc Ω. It is seen that rules (Ref. [4, 7, 8, 9, 12, 15] ) meant for the numerical integration of the integral = ∫ () − 0 ; lead to uncontrolled instability when those are applied for the approximation of the integral given in equation (1). This is due to the presence of singular point  0 of order >1 on the path of integration . The integral defined in equation (1) is called as hyper singular integral in complex plane. The study of its real counter part has been going on for a long time and has been documented in a number of publications (Ref . [6, 13, [?], 17-22] ).  ∗ =∫   () (−) 2 ;  <  < . (2) However, a very few rules in the course of numerical integration have devised for the former. Therefore, in this study we have proposed a numerical scheme for the numerical computation of the integral given in equation (1). 2. Description of the scheme for numerical evaluation of complex hyper singular integral To establish the scheme for the numerical computation of the hyper singular integral (,  0 )=∫  () ( −  0 )  ;  ∈ ℕ − {1}; we assume here that the function () is analytic and infinitely differentiable on the disc Ω = { ∈ ℂ: | −  0 | <  = |ℎ|, > 1}.