| Summary: | In this paper, a fast and accurate numerical Clenshaw-Curtis quadrature is proposed for the approximation of highly oscillatory integrals with Cauchy and logarithmic singularities, <inline-formula> <math display="inline"> <semantics> <mrow> <msubsup> <mi>⨍</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mn>1</mn> </msubsup> <mfrac> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mi>log</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>−</mo> <mi>α</mi> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow> <mi>i</mi> <mi>k</mi> <mi>x</mi> </mrow> </msup> </mrow> <mrow> <mi>x</mi> <mo>−</mo> <mi>t</mi> </mrow> </mfrac> <mi>d</mi> <mi>x</mi> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>∉</mo> <mo>(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> <mo>,</mo> <mi>α</mi> <mo>∈</mo> <mo>[</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>]</mo> </mrow> </semantics> </math> </inline-formula> for a smooth function <inline-formula> <math display="inline"> <semantics> <mrow> <mi>f</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>. This method consists of evaluation of the modified moments by stable recurrence relation and Cauchy kernel is solved by steepest descent method that transforms the oscillatory integral into the sum of line integrals. Later theoretical analysis and high accuracy of the method is illustrated by some examples.
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