See also the subsection on Hankel functions below.
2.
The Hankel functions admit the following integral representations for Re ( " x " ) > 0:
3.
The importance of Hankel functions of the first and second kind lies more in theoretical development rather than in application.
4.
In this domain total field is written in terms of Bessel and Hankel function solutions to the cylindrical Helmholtz equation.
5.
The Bessel function of the second kind then can be thought to naturally appear as the imaginary part of the Hankel functions.
6.
The spherical Hankel functions appear in problems involving spherical wave propagation, for example in the multipole expansion of the electromagnetic field.
7.
The radial part of this wave function consists solely of the spherical Bessel function, which can be rewritten as a sum of two spherical Hankel functions:
8.
The Hankel functions are used to express outward-and inward-propagating cylindrical wave solutions of the cylindrical wave equation, respectively ( or vice versa, depending on the sign convention for the frequency ).
9.
This, along with third constraint, selects Hankel function of the first kind as the only converging solution at infinity ( the singularity at the origin of these functions does not matter since we are now outside the sphere ):
10.
The Hankel functions of the first and second kind are used to express outward-and inward-propagating cylindrical wave solutions of the cylindrical wave equation, respectively ( or vice versa, depending on the sign convention for the frequency ).
मोबाइल