On the theory of the Kelvin ship-wave source: the near-field convergent expansion of an integral

The velocity potential of the Kelvin ship-wave source is fundamental in the mathematical theory of the wave resistance of ships, but is difficult to evaluate numerically. We shall be concerned with the integral term F ( x, ρ, ∝ ) = ∫ ∞ -∞ exp {— 1/2 ρ cosh (2 u — i ∝ )} cos ( x cosh u )d u in the source potential, where x and ρ are positive and —1/2 π ≼ ∝ ≼ 1/2 π , which is difficult to evaluate when x and ρ are small. It will be shown here that F ( x, ρ, ∝ ) = 1/2ƒ( x, ρ, ∝ ) + 1/2ƒ( x, ρ, ─∝ ) + 1/2ƒ( ─x , ρ, ∝ ) + ½ƒ( ─x, ρ, ─∝ ), where ƒ( x, ρ, ∝ ) = P 0 ( x, ρ e -i ∝ ) Σ g m ( x, ρ e i ∝ ) c m ( x, ρ e -i ∝ ) + P 1 ( x, ρ e -i ∝ ) Σ g m ( x, ρ e i ∝ ) b m ( x, ρ e -i ∝ ) + Σ g m ( x, ρ e i ∝ ) a m ( x, ρ e -i ∝ ) In this expression each of the functions g m ( x, ρ e i ∝ ), a m ( x, ρ e -i ∝ ), b m ( x, ρ e -i ∝ ), c m ( x, ρ e -i ∝ ), satisfies a simple three-term recurrence relation and tends rapidly to 0 for small x and ρ when m → ∞, and the functions P 0 ( x, ρ e -i ∝ ) and P 1 ( x, ρ e i ∝ ) are simply related to the parabolic cylinder functions D v (ζ) respectively, where ζ = — i x (2 ρ ) -1/2 e 1/2 i ∝ .