The double exponential formula for oscillatory functions over the half infinite interval

The double exponential formula is known to be very powerful for evaluation of various kinds of integrals, in particular integrals with end point singularities or integrals over the half infinite interval. It is also known that a weak point of this formula is the inefficiency when applied to a slowly decaying oscillatory integral over the half infinite interval such as I = ∫0∞f1(x) sin x dx, f1(x) is an algebraic function. In this paper we propose a new type of the double exponential formula which is quite efficient for evaluation of the integral mentioned above. It is based on such a transformation that makes the points of the formula after the transformation approach to the zeros of sin x double exponentially for large x.