The Convergence of the Tangential Scattered Magnetic Field Based on the Vertical Improper Integral

The tangential scattered magnetic field of a perfectly electric conducting surface is usually expressed as an improper integral. However, the improper integral is divergent. Hence, the field is expressed as a limit of an integral in some references. In this article, the limit of the integral is called a vertical improper integral. A stronger smooth condition of the surface is added to ensure the convergence of the vertical improper integral. The relationship between the vertical improper integral and the Cauchy principle value integral is established. An example shows that the tangential scattered magnetic field is infinite at a first-order smooth point if it is assumed that the surface current density is non-zero at the point. By the boundary condition, it is obtained that the surface current density should be zero at the point. A numerical example is given to demonstrate the effectiveness of the vertical improper integral method. The numerical result also shows that singular integrals need not be calculated when the magnetic field integral equation is computed based on the method of moments.