Advances in Theoretical and Applied Mathematics ISSN 0973-4554 Volume 7, Number 1 (2012), pp. 59-71 © Research India Publications http://www.ripublication.com/atam.htm
Construction of Newton – Cotes Types of Rules for Approximate Evaluation of Real Cauchy Principal Value of Integrals. R.N. Das and M.K. Hota* School of Mathematics & Computing Gandhi Institute for Technological Advancement, Bhubaneswar, India *Corresponding Author E mail:
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Abstract Newton-Cotes types of rules have been constructed for approximate evaluation of Cauchy principal value integrals and the error bound have been obtained. The rules have been numerically verified in case of certain Cauchy principal value integrals. Keywords: Cauchy principal value, degree of precision, error bound, cauchyinequality, error constant.
Introduction A definite integral I(f ) =
b
∫ f ( x ) dx a
is defined as a singular integral if : i. the interval of integration a ≤ x ≤ b is either right infinite or left infinite or of double infinite or ii. the integrand f ( x ) is unbounded at an interior point ' c ' in the range of integration including one or both points of integrations. or iii. in many cases the range of integration is infinite and the function is also unbounded in the range of integration. Such integrals are also called
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R.N. Das and M.K. Hota singular integrals. However these types of singular integrals can be expressed as sum of integrals of the type (i) and (ii) by choosing a suitable point in the range of integration.
In a broader sense, in numerical analysis an integral is defined to be a singular integral if standard quadrature rules such as: i. Newton-cotes rules ii. Gauss rules when applied to these integrals for approximate evaluation fails to yield values having desired accuracy. This definition of singular integral is due to Atkinson [1]. The definition is highly appreciated and is justified as there are integrals although analytically not singular but found to be ill behaved in numerical integration by standard quadrature rules. e.g. the integral:
I( f ) =
1
dx
∫ 2 −1 1+ x
is not numerically integrated accurately by any of the standard quadrature rules although it is a mathematically nonsingular integral. Integration rules have been constructed for approximate evaluation of the singular integrals of the types: (a) (b)
I( f ) = I( f ) =
∞
∫−∞ e ∞
∫e −∞
−x
− x2
f ( x ) dx f ( x ) dx
which are respectively known as Gauss-Laguerre rule and Gauss- Lobatto rule (ref [1]). Singular integrals of the type (ii) occur frequently in many branches of physics, in the theory of aerodynamics and scattering theory e.t.c. In particular the integrals of the type:
I ( f ,c) =
f ( x) ∫ x − c dx ; b
a
