## Abstract

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,$\rho $,$\alpha $) = $\int_{-\infty}^{\infty}$ exp{-$\frac{1}{2}\rho $ cosh (2u - i$\alpha $)} cos (x cosh u) du in the source potential, where x and $\rho $ are positive and -$\frac{1}{2}\pi \leq \alpha \leq \frac{1}{2}\pi $, which is difficult to evaluate when x and $\rho $ are small. It will be shown here that F(x,$\rho $,$\alpha $) = $\frac{1}{2}$f(x,$\rho $,$\alpha $) + $\frac{1}{2}$f(x,$\rho $,-$\alpha $)+$\frac{1}{2}$f(-x,$\rho $,$\alpha $)+$\frac{1}{2}$f(-x,$\rho $,-$\alpha $), where f(x,$\rho $,$\alpha $) = P$_{0}$(x,$\rho $ e$^{-\text{i}\alpha}$) $\sum $g$_{m}$(x,$\rho $ e$^{\text{i}\alpha}$) c$_{m}$(x,$\rho $ e$^{-\text{i}}\alpha $) + P$_{1}$(x, $\rho $ e$^{-\text{i}\alpha}$) $\sum $g$_{m}$(x,$\rho $ e$^{\text{i}\alpha}$)b$_{m}$(x,$\rho $ e$^{-\text{i}\alpha}$) + $\sum $g$_{m}$(x,$\rho $ e$^{\text{i}\alpha}$)a$_{m}$(x,$\rho $ e$^{-\text{i}\alpha}$). In this expression each of the functions g$_{m}$(x,$\rho $ e$^{\text{i}\alpha}$), a$_{m}$(x,$\rho $ e$^{-\text{i}\alpha}$), b$_{m}$(x,$\rho $ e$^{-\text{i}\alpha}$), c$_{m}$(x,$\rho $ e$^{-\text{i}\alpha}$), satisfies a simple three-term recurrence relation and tends rapidly to 0 for small x and $\rho $ when m $\rightarrow \infty $, and the functions P$_{0}$(x, $\rho $ e$^{-\text{i}\alpha}$) and P$_{1}$(x,$\rho $ e$^{-\text{i}\alpha}$) are simply related to the parabolic cylinder functions D$_{\nu}$($\zeta $) respectively, where $\zeta $ = -ix(2$\rho $)$^{-\frac{1}{2}}$e$^{\frac{1}{2}\text{i}\alpha}$.

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