Hankel–Bessel Ladder Identities Claim. For suitable \(f_m(r)\) and \(k>0\), \[ \mathscr{H}_{m+1}\!\left[\left(\partial_r-\frac{m}{r}\right) f_m\right](k) =-\,k\,\hat{f}_m(k). \tag{1} \] Here \(\hat f_m(k):=\mathscr H_m[f_m](k)=\displaystyle\int_0^\infty f_m(r)\,J_m(kr)\,r\,dr\).

basic concept ninja ninja is a build system—a tool that actually runs the compile and link commands for a software project. T...

The case in which the water has infinite depth is considered, the effect of surface tension as well.

This is the notes for Chapter 5.4, mainly about ship waves.

The article analyzes ship wave patterns in deep water with surface tension, showing that each wave crest corresponds to a con...

The article develops a mathematical model for one-dimensional gravity-capillary ship waves, using residue theory to analyze w...

在英文数学论文中，推导公式时连接前后式子的常用连接词和短语可以分为以下几类，以确保逻辑清晰、推导严谨： 1. 表示直接推导或结果 连接词/短语： Therefore Example: "Therefore, Equation (3) can b...

In dispersive water waves (e.g., surface gravity waves on deep or shallow water), several concepts describe how waves travel,...

paper core

Pressure to the surface to a stream