Bessel functions orthogonality

The boundary conditions (4.39), the orthogonality of the vector harmonics, and the form of the expansion of the incident field dictate the form of the expansions for the scattered field and the field inside the sphere the coefficients in these expansions vanish for all m = = 1. Finiteness at the origin requires that we take y (kjr), where kj is the wave number in the sphere, as the appropriate spherical Bessel functions in the generating functions for the vector harmonics inside the sphere. Thus, the expansion of the field (Ej,H,) is... 

An extensive study of analytical techniques used in conduction heat transfer requires a background in the theory of orthogonal functions. Fourier series are one example of orthogonal functions, as are Bessel functions and other special functions applicable to different geometries and boundary conditions. The interested reader may consult one or more of the conduction heat-transfer texts listed in the references for further information on the subject. 

For continuum channels the radial orbitals in (7.140) are not bounded and the integral is divergent. The choice of radial functions in (7.136) must be based on intuition obtained from a study of ionisation, which is treated in chapter 10. A necessary condition for a reasonable distorted wave in the distorted-wave Bom approximation for ionisation is that it should be orthogonal to the initial state in the ionisation amplitude. For computational simplicity we set Fop,7l equal to zero and orthogonalise the resulting Ricatti—Bessel functions to all the states of P space using (5.83). 

The augmented Bessel function defined through (5.19,21,23) has several desirable properties. It is energy independent, it is everywhere continuous and differentiable, it is orthogonal to the core states of its own muffin-tin well as shown by (3.23), and it is finally proportional to the function (D j, r). 

The derivation of the solution and mathematical details are reported in the separate study Claesson (2003). The difference l-tT has homogeneous boundary conditions. Multiplication with r° removes the factor la in (10). The method of separation of variables gives Bessel functions as solution. The eigenfunctions w (r) are orthogonal, and the initial condition is satisfied by a suitable choice of coefficients A . We have ... 

For Bessel functions and other types of special functions to be introduced later, it is possible to construct orthonormal sets of basis functions (/>n(x), which satisfy the orthogonality and normalization conditions ... 

The final step is to choose the A so that w(r, 7) satisfies the initial condition 777(7, 0) = -(1 - r2)/4. The general Sturm-Louiville theory16 guarantees that the eigenfunctions (3-106) form a complete set of orthogonal functions. Thus it is possible to express the smooth initial condition (1 -r2) by means of the Fourier-Bessel series (3-109) with 7 = 0, that is,... 

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