Fast and stable computation of highly oscillatory and/or exponentially decaying integrals using a Clenshaw-Curtis product-integration rule

We propose, analyze, and implement a quadrature method for evaluating integrals of the form $$\int _0^2 f(s)\exp (zs)\, \textrm{d}s$$ , where z is a complex number with a possibly large negative real part. The integrand may exhibit exponential decay, highly oscillatory behavior, or both simultaneously, making standard quadrature rules computationally expensive. Our approach is based on a Clenshaw-Curtis product-integration rule: the smooth part of the integrand is interpolated using a polynomial at Chebyshev nodes, and the resulting integral is computed exactly. We analyze the convergence of the method with respect to both the number of nodes and the parameter z. Additionally, we provide a stable and efficient implementation whose computational cost is essentially independent of z and scales linearly with the number of Chebyshev nodes. Notably, our approach avoids the use of special functions, enhancing its numerical robustness.