Transient evolution of a photon gas in the nonlinear QED vacuum Page: 10 of 21
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where an are polynomials in y.
By inspecting equation (12), it is clear that special attention must be paid to the boundaries of the
integration domain. The y integral in equation (15) can be approached by first dividing it into
two disparate cases. When q < k, the integration region covers the entire interval y E [-1,1];
when q > k, however, the integration region shrinks to y E [1, , where the lower bound is
2k 2k 2k2
y, =1+ - - . Integration over y can now be performed in terms of the incomplete beta
h q hq
function, B(x;a,b) [13]:
dy = y;'"-" B ] ;1+m, -n -B -;1+m, -n (16)
Y y0 -y) _ + y0 y0
with yo = -+- .
2qh
For q < k, after integration over y, the functionf itself can be further subdivided into two regions
based on the value of h:
= 32q 2228h2 +15hk+8k2 -q7 h+k(2h2 +3hk+2k2 +7 h2 +hk+k2 2-9qs h+k +2q q<h (17)
35h2k2 ) q q) , q<k
32 q (2h+9h3k+q2 (16hz+35hk+21k2)-q(h+k)(9hz+21hk+14k2)+16h2kz+14hk3-14 a h+k +7k+7 , <q (18)
This second division is a reflection of symmetry in the problem. Other than the order of
integration, neither h nor k is special; hence any feature in the domain of one variable must be
reflected in the domain of the other.
Similarly, for the case when q > k, the expression after integration over y divides into twosubdomains:
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Wu, S Q & Hartemann, F V. Transient evolution of a photon gas in the nonlinear QED vacuum, report, October 4, 2011; Livermore, California. (https://digital.library.unt.edu/ark:/67531/metadc834710/m1/10/: accessed July 16, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.