Bessel functions properties

Bessel functions have many interesting properties that will be presented here without proof, e.g. the recursion formula... 

For properties of Bessel functions, see for example, Walas, Modelling with Differential Equations in Chemical Engineering, 1991). Applying the condition at the center, Eq (4),... 

The result (03.5) is very useful for deducing properties of the modified Bessel function In x) from those of the Bessel function For instance, when n is an integer... 

The solution to this problem is to transform, or half-transform, the S matrix from the body-fixed to the space-fixed axis system then to use the known analytic properties of the spherical Bessel functions, which are the solutions to the potential-free scattering problem in the space-fixed axes and finally to transform back to the body-fixed axes and then to use Eq. (4.46) to calculate the differential cross section. 

The integral in Eq. (3.34) is bir >mm due to the orthonormal property of the spherical harmonics. The second line is the Wronskian of the spherical modified Bessel functions. 

The key of the proof is the properties of the spherical modified Bessel function of the first kind, For small values of u, the function ii u) has... 

Actually, by starting with Equations (C.8) and (C.9) as the definitions, all the properties of the spherieal modified Bessel functions can be obtained, without tracing back to the formal definition. Equations (C.6) and (C.7). 

It follows from the properties of the Bessel functions that... 

The properties of the modified spherical Bessel functions (22) and Equation 33... 

New functions are sometimes defined as a solution to differential equation, and simply named after the differential equation itself. It is the purview of the mathematician to understand the properties of these functions so that they can be used confidently in numerous other applications. The Bessel function is of this kind, the solution of a differential equation that occurs in many applications of engineering and physics, including heat transfer. 

In the box, two additional methods to obtain Bessel functions are summarized. The generating function relates Bessel functions to the exponential, Spiegel, 1971[3]. This relation is useful for obtaining properties of the Bessel function for integral n. Recursions of the Bessel functions are generally derived this way. Bessel s integral relates Bessel and trigonometric function. 

In this appendix, the essential properties of Bessel functions that are required in physical applications have been discussed. There are many books and articles on Bessel functions, and tables and graphs of their values and properties. There are also several good books giving the essentials of Bessel functions for scientists and engineers. Every textbook on hydrodynamics, elasticity, electromagnetism and vibrations will have examples of the use of these functions. Bowman, 1958 [4], is recommended. 

To solve equation (18), subject to the boundary conditions given in equations (19)-(21) and to the additional requirement that y be bounded, is a straightforward mathematical problem. By the method of separation of variables, for example, it can be shown (through the use of recurrence formulae and other known properties of Bessel functions of the first kind) that... 

The interesting case of potentials with tails falling off like 1 /i was recently treated by Mortiz et al. [59]. They reported many aspects of the near-threshold properties for these potentials. However, they did not give values of the critical parameters Xc and a. There is no general solution of the Schrodinger equation for power-law potentials, but for tails going to zero faster than 1/r2 and E(XC) 0, the solutions are Bessel functions [60] that can be normalized for... 

After some further manipulation, using known properties of the modified Bessel functions and putting in the definition of a, we can write this as... 

The Hankel transform may be used to solve numerous boundary-value problems in a relatively straightforward way, using various properties of Bessel functions. We present here the solution to one of the boundary-value problems (Davies, 2002) of interest. 

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 linear stability characteristics of the jet are specified by Eq. (10.4.32), where we note that (3 alpa, which may be compared with the plane capillary wave result where crlpX. This behavior is not surprising and can be deduced from dimensional arguments. Indeed, for the jet when a 1, that is, when the wavelengths are small compared with the jet radius, we have from the properties of the Bessel function that /(,( )/I (a) = 1. With f3 = io), Eq. (10.4.32) reduces to the dispersion relation o) - k crlp for stable, sustained surface capillary waves on deep water (Eq. 10.4.19). 

Most of the properties of the modified Bessel function may be deduced from the known properties of J (x) by use of these relations and those previously given. 

Let us now introduce p via /3(/3 + 1) = a. Weakly attractive potentials correspond to -1/2 < /I < 0 and repulsive potentials to > 0. In both cases, the solution of the Schrodinger equation at large r can be written as a linear combination of the Ricatti-Bessel functions /(kr) and h/s(kr), and the order of the corresponding cylinder functions v = p + 1/2 remains positive. Using analytic properties of the Bessel functions as kr 0, to lowest order in k,... 

Eq. (17.65) detemiines the stability conditions for the jet. It should be noted that fP Llpa, whereas in the plane case involving capillary waves at the surface of a deep reservoir we had aP according to (17.52). These perturbations also could be derived from (17.73). Indeed, consider the case of a = ak 1 or a 2, i.e. perturbations whose wavelength is small in comparison with the jet s radius. It then follows from the properties of the Bessel function that In ct) 1. Putting p = ico into (17.65), one finds that aP = k L/p L/pX, which corresponds to neutrally stable capillary waves on the surface of a deep water reservoir. 

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