Jacobian elliptical function

Theta functions are special functions related to Jacobian elliptic functions (Morse and Feshbach, 1953 Widder, 1975) with special properties that make then extremely useful to calculate solutions to diffusion problems for small values of time. Three of the four theta functions will be used in the present context... 

In this chapter we shall describe the procedure for expressing the phase-integral formulas derived in Chapter 5 in terms of complete elliptic integrals. The integral in question is first expressed in terms of a Jacobian elliptic function and then in terms of complete elliptic integrals. Different elliptic functions are appropriate for different phase-integrals. For practical calculations it is most convenient to work with real quantities. For the phase-integrals associated with the r -equation it is therefore appropriate to use different formulas for the sub-barrier case and for the super-barrier case. We indicate in this chapter, where we use the notations L 2n+l L 2n+1 K< 2n+1 ) instead of the notations 2 , 2n used previously, the main... 

Considering first L in the sub-barrier case in Fig. 5.1b, we introduce the Jacobian elliptic function snu by the transformation... 

Recalling (5.15a,b) and (5.7a), and using (6.1), (6.3) and general properties of Jacobian elliptic functions, we obtain after some calculations... 

Concerning the spheroconal coordinates expressed through Jacobian elliptic functions used in Ref. [3], its Eq. (50) is reproduced here... 

Mathematically, the classical problem is completely soluble in terms of Jacobian elliptic functions, the quantum problem in terms of Mathieu, or elliptical cylinder, functions. The limiting behaviour of quantum solutions for the two modes are extremely well known, and the literature contains extensive discussions of these phenomena in the limiting cases both can be handled by perturbation techniques [11, ... 

Analytical studies of Eq. (168) have used different changes of variables to write the electric field in terms of either Painleve transcendents [68] or Jacobian elliptic functions [88]. Alternatively, asymptotic expansions have also been used [68, 87, 89, 90]. The approximate solution methods have neglected different terms of Eq. (168). Thus, while Urtenov and Nikonenko [55, 91, 92] have considered that the space charge density is quasiuniform, dSi/dx 0, Bass [93] assumed that eE RTand I/II 8/Lo to ex-... 

Here cn t, sn t, dn t are standard Jacobian elliptic functions corresponding to an elliptic curve with the parameter... 

The above formulae and the properties of Jacobian elliptic functions imply that the coordinates Mi are doubly periodic functions of complex time t which have in the rectangle of periods poles of order one at points A, ByC D with residues... 

In order to transform the wavemaker geometry to a Jacobian elliptic function, it must be rotated and translated as shown in Fig. 2.11. The 90° rotation to the z -plane is given by... 

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