(&osh2aoz + ach2oioz)]                              (17)
exp (-az)
R2 =    4a   ) (ch2cxoz - 1)                                  (18)
R3 =         1 + exp(-2az) sh2-o 1 -   _   ch2z -    2       (19)
40    \   aof)
Here
ao =   a2 - Q2                         (20)
If ao is imaginary, then
chaoz -* cosw1z
shcypz  _  sin w, z  w1 =   SZ2 - a2             (21)
ao          wl
APPLICATIONS TO LONGITUDINAL COOLING
In this case y = z - zs. (zs is a coordinate of equilibrium particle),
d     dE     P
2a-mc232y   dz ion Lo+ dK A/fA
d [(d)      v] - 2a1+2a2+23                 (22)
The first term in RHS describes longitudinal cooling due to wedges, the second
one damping due to change of longitudinal mass, and the third one - so
named "natural" cooling due to slope of the ionization losses curve [2]. Here
o=     _i"                           (23)
dX (dE' io
We see, that if (d)  does not depend on X, the derivative is equal to zero,
and the first term in (22) disappears. Where X is a transverse coordinate.
Parameter A is defined by
A =R     +                             (24)
R is a radius of curvature for ideal trajectory. Q is the synchrotron frequency
and in linear approximation may be written as
27rEac cos (27 zs) A
mc2A,32'                          (25)