Abstract
In this paper, we discuss the integrals of oscillatory kind function with Cauchy principal value in point zero which have the form like , where f (x) is smooth function and r is odd integer. In this integral, xr has several stationary points , and the Cauchy principal value . We use some integral technique to transform it into the form like so that we can calculate it. At the end, we give some numerical examples to prove the accuracy of this method.
Highlights
In this paper, we concerned about the Cauchy principal value in oscillatory kind function which g ( x) = xr, have the form like ( ) b f xr eiωxr∫ a x dx where f ( x) is smooth and ω ∈ R, generally ω is much large
( ) b f xr eiωxr principal value in point zero which have the form like ∫a x dx, where f ( x) is smooth function and r is odd integer
We concerned about the Cauchy principal value in oscillatory kind function which g ( x) = xr, have the form like
Summary
Levin method to solve the integral with Cauchy principal value. It is a great idea for us to solve Cauchy principal value problem and we are using this idea to solve problem in this paper. We take his idea of choose path, which make it easy to have a good path to integral. With the function g ( x) not smooth, maybe has stationary points and so on, they use Gauss-Fraud quadrature to solve the path near the stationary points. We firstly transformed the equation into the form which we known before, we take the path to get the solution with Gauss-Laguerre quadrature.
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