Асимптотическое разложение интеграла с двумя параметрами
A classic pursuit problem is studied in which two material points - a Pursuer and a Pursued - move in a plane at constant speeds. The velocity vector of the Pursued does not change its direction and the velocity vector of the Pursuer turns and always aims at the Pursued. If the Pursuer moves at a higher speed, it will overtake the Pursued for any initial angle between velocity vectors. For example, a crane system simultaneously producing three movements: rotation, extension/retraction and luffing of the telescopic boom may seize the load moving uniformly in a straight line while the crane is standing motionless. In the coordinate system associated with the Pursuer, the length of the Pursued's trajectory is given by an integral, depending on two parameters: the ratio of the initial velocities of two points and the initial angle between them. The theorem on the asymptotic expansion of this integral Is formulated and proved under the assumption that the speed of the Pursuer is much greater than the speed of the Pursued. The first two nonzero terms of the asymptotic expansion provide fast convergence to the exact value of the integral because of the absence of the first- and the third-order terms of the asymptotics. The third nonzero term of the fifth order allows to determine the difference of trajectory lengths corresponding to the adjacent initial angles between the velocities of the points.