Understanding and Prediction of Ship Motions 





K(cd) = I K(t) e ''^'dt = K^(a;) - LK^(aj). 



Also, 



K(-a)) = r K(t)e'''*dt = K^(oj) + iK^(aj) ^ K(co) .* 



These relations depend quite explicitly on the condition that K(t) = for t < 0; 

 this has frequently been described as an effect of "causality," i.e., the system 

 does not respond before t = if the disturbance comes at t = 0. 



If now we consider ^ as a complex variable, it is clear from the definition 

 of the Fourier integral K(co) that the convergence of the integral can only be 

 quickened when Im w < o. Then K(«) cannot have any singularities in the lower 

 half of the w-plane. But for an analytic function we can use Cauchy's integral 

 formula: 



if ^(^')d^' 



KfO)) = — r ; 



^ ' 2771 J OJ' - CO 



See the figure. 



If K(m) vanishes far from the origin in the lower half -plane (no matter how 

 slowly), the contribution from the semi-circle vanishes as the radius grows to 

 infinity, and so we can replace the integral over the closed contour C by a con- 

 tour integral along the real axis from -oo to +oo. 



Now we let oj approach the real axis (from below), and we indent the contour 

 above the real axis, so that for real co 



K(a)) = 



^'ni^ 



K(a)')dw' 



77i K(oS) 



^^ I 



K(aj')daj' 



*The long bar denotes the complex conjugate quantity. 



79 



