Bessel functions modified

In Eq. (26), M is the hydrogen mass, X labels the mode, is the atomic eigenvector for hydrogen / in mode X, and co, is the mode angular frequency. is the number of quanta of energy Ao>, exchanged between the neutron and mode X. is a modified Bessel function. 

The analytical solution of this equation Is Known (9) (10) In terms of modified Bessel functions of the first kind. AccorxUngly, the dlstrltutlon of the active chains In the particles with volume V, fn(v)/f(v), and the average nunher of active chains In the same — 00... 

In Eq. (77), x = h(o/2kg T is the reduced internal frequency, q = EJhoi the reduced solvent reorganization energy, p = hElha> the reduced electronic energy gap and / (z) the modified Bessel function of order m. The quantity S is a coupling parameter which defines the contribution of the change in the internal normal mode ... 

Expanding each of the exponential factors in a series of modified Bessel functions, the MaxEnt distribution can be written ... 

This expression can be put in terms of modified Bessel functions. Applying the method of steepest descent to Eq. (3.52) yields... 

In this equation, Jv is the Bessel function of the first kind, and Kv is the modified Bessel function of the second kind, U = aikfnf—fi1)112, W = a(f21—konf)if2,... 

The Modified Bessel Functions. By an argument similar to that employed in 1 we can readily show that Laplace s equation in cylindrical coordinates d2yi, 1 dy> 1 d tp d2yi... 

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... 

In this Chapter, we present step-by-step derivations of the explicit expressions for matrix elements based on the spherical-harmonic expansion of the tip wavefunction in the gap region. The result — derivative rule is extremely simple and intuitively understandable. Two independent proofs are presented. The mathematical tool for the derivation is the spherical modified Bessel functions, which are probably the simplest of all Bessel functions. A concise summary about them is included in Appendix C. 

The standard linear-independent solutions for Eq. (3.4) are the spherical modified Bessel functions, )(m) and ki(u) (Arfken, 1968). A brief introduction to them is provided in Appendix C. These so-called special functions are actually elementary functions. ... 

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... 

This is the modified Bessel equation of order v = n + Vi. The solutions of Eq. (C.3) are modified Bessel functions of the first kind, which is defined through the Bessel function Jfx) as... 

---