For the spherical Bessel functions the orthogonality relation is:
2.
For large, the spherical Bessel functions decay as, giving the radiated field above.
3.
:Expansion in spherical Bessel functions would be more appropriate for domains of this kind.
4.
Thus, we use the zeros of the spherical Bessel functions to find the energy spectrum and wavefunctions.
5.
The three-dimensional solutions of the Helmholtz Equation can be expressed as expansions in spherical harmonics with coefficients proportional to the spherical Bessel functions.
6.
The terms in this expansion are spherical harmonics ( which give the angular dependence ) multiplied by spherical Bessel functions ( which give the radial dependence ).
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.
And since sin y / y is the zeroth spherical Bessel function, the above would involve the sph bess function along the imaginary axis, K _ 0.
9.
In fact, there are simple closed-form expressions for the Bessel functions of half-integer order in terms of the standard trigonometric functions, and therefore for the spherical Bessel functions.
10.
The spherical bessel function j _ \ ell ( kr') can also be simplified by assuming that the radiation length scale is much larger than the source length scale, which is true for most antennas.
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