Properties of Bessel Functions

It is easily verified that all Power Series presented thus far as definitions of Bessel functions are convergent for finite values of x. However, because of the appearance of ln(A ) in the second solutions, only JpM and Ip(x) are finite at jc = 0 (p 0). Thus, near the origin, we have the important results  

The sign in the last expression depends on the sign of F(m p 1), as noted in Eqs. 3.149 and 3.161. However, it is sufficient to know that a discontinuity exists at X = 0 in order to evaluate the constant of integration. We also observed earlier that ln(jc) appeared in the second solutions, so it is useful to know (e.g.. Example 3.5) 

Asymptotic expressions are also useful in taking limits or in finding approximate solutions for small values of x, the approximations are 

For large arguments, the modified functions sustain exponential type behavior and become independent of order (p may be integer or zero)  

However, for large arguments, Jpix) and Yp(x) behave in a transcendental manner  

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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),... 

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 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. 

We were unable to express net heat flux for the Pin-promoters in Example 3.5, in the absence of differential properties of Bessel functions. Thus, the temperature profile obtained in Eq. 3.175 was found to be expressible as a first order, modified Bessel function... 

Again, the multiplicative constant for K is taken as unity. The properties of Bessel function are discussed in Chapter 3. 

Again using the properties of Bessel Functions (Wylie, 1960), we get... 

Using the properties of Bessel functions, i/ (x) can be expressed in terms of Airy functions Ai of negative and positive arguments, respectively [153]. Then, we obtain the uniformly valid Langer solution [48, 64] ... 

From the properties of Bessel functions, the integral in equation (6.6) is known to be equal to 0.216i . Hence,... 

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. 

The additional properties of these functions may all be derived from the above relations and the known properties of ] x). 12. Complete solutions to Bessel s equation may be written as... 

We need the differential or integral properties for Bessel functions to compute the net rate of heat transfer. We discuss these properties in the next section, and then use them to complete the above example. 

The spatial domain for problems normally encountered in chemical engineering are usually composed of rectangular, cylindrical, or spherical coordinates. Linear problems having these types of domain usually result in ODEs (after the application of separation of variables) that are solvable. Solutions of these ODEs normally take the form of trigonometric, hyperbolic, Bessel, and so forth. Among special functions, these three are familiar to engineers because they arise so frequently. They are widely tabulated in handbooks, for example, the handbook by Abramowitz and Stegun (1964) provides an excellent resource on the properties of special functions. 

Figure 12.3b shows the widths of the Bragg layers as a function of the layer number of the CBNL depicted in Fig. 12.3a. There are two notable properties of the Bragg layers (1) the width of the high-index layers is smaller than the width of the low-index layers, and (2) the width of the layers decreases exponentially as a function of the radius, converging asymptotically to a constant value. The first property exists in conventional DBRs as well and stems from the dependence of the spatial oscillation period, or the wavelength, on the index of refraction. The second property is unique to the cylindrical geometry and arises from the nonperiodic nature of the solutions of the wave equation (Bessel or Hankel functions) in this geometry.

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... 

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. 

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... 

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