Hyperbolic Bessel functions

B = magnetoelastic coupling E = Young s modulus H = magnetic field //+I/2 = hyperbolic Bessel function J = quantum number... 

Ki denotes the lowest order anisotropy constant for cubic crystals defined in section 4 A denotes any of the lowest order magnetostriction constants As, Aioo or A i. /,+i/2 is a reduced hyperbolic Bessel function of order /, and i " [mR]... 

For the important case of odd values of 8, corresponding to three-dimensional l waves l = 0,1,2,... for 8 = 3,5,7,..., the Bessel functions reduce to the usual trigonometric and hyperbolic functions times powers of r [55]. In this case, everything but the solution of the transcendental equation for the energy can be done analytically. The advantage is that we can go further to very large values of R, and the convergence for R oc could be studied without any extrapolation process. 

It is assumed that the exponential and logarithmic functions are thoroughly familiar to the reader as also are their relatives the circular and hyperbolic functions. However, the Bessel functions are introduced to the student much later and have less claim to familiarity. The exponential integral is also a function which occurs in several places and is worthy of some explanation. This appendix is therefore intended to provide a little background on the applications of these functions. 

The convergence of this series is assured for all r and the function so defined is called the modified Bessel function of zeroth order. It is tabulated just as is the hyperbolic cosine the two are shown in Fig. A. 1. 

During an experiment the sample is cold, ca 20K (= 14 cm ). The lowest internal vibrations are typically about 300 cm and the hyperbolic sine function will, except at the very lowest energies, have an argument greater than ten. The argument of the Bessel function is, therefore, less than 10 and it can be safely represented by the first term of its power series expansion. Where, for an arbitrary argument x ... 

Expand the Bessel function in terms of the hyperbolic functions sink and cosh ... 

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. 

To facilitate the solution of electroosmotic velocity for arbitrary values of fluid behavior index, one may approximate the hyperbolic sine function and the first-order modified Bessel function of the first kind, respectively, as [6]... 

In these expressions, the relations (5.215) have been used as the changing of variables, along the connection between the hyperbolic functions and the series of the modified Bessel functions while for the integrals calcula-... 

The solutions to these equations have already been discussed in much detail. The fi equation yields for Z the trigonometric and ordinary Bessel functions, and the X equation, the hyperbolic and modified Bessel functions. 

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