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We show how the zeroth-order Hankel transform diagonalises the radial bi-Laplacian, clarify the sign convention \(\nabla^4 \leftrightarrow k^4\), and then apply the same transform machinery to an axisymmetric linear free-surface problem to obtain explicit representations for \(\eta(r,t)\) and \(\hat w(s,z,t)\).

We collect a few basic identities for the gamma and polygamma functions, their link to the Hurwitz zeta function, standard asymptotics for Bessel functions, and a useful Bessel product expansion and inequality for \(J_n(z)\).

the c++ casting operators the four c++ casting operators are static_cast dynamic_cast reinterpret_cast const_cast use stati...

move constructor and move assignment operator Take a look at the addition operator + as implemented in demolist12_4. Notice t...