The Stokes Phenomenon, Borel Summation and Mellin-Barnes Regularisation

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The Stokes phenomenon refers to the emergence of jump discontinuities in asymptotic expansions at specific rays in the complex plane. This book presents a radical theory for the phenomenon by ...
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Generalised Terminants

Pp. 133-153 (21)

Victor Kowalenko


Generalised terminants are produced when the coefficients of the two types of series considered in Ch. 7 are set equal toΓ(pk+q), where pandqare both real and positive and the variablez is altered to z β , where βcan be much greater than unity. Ch. 9 is concerned with the derivation of the MB-regularised forms for the regularised values of both types of generalised terminants over the entire complex plane, which are presented in Propositions 4 and 5. These results are then simplified by considering special cases of p, the first where it is the reciprocal of a natural natural number and the second, where it equals 2. The chapter concludes by evaluating the regularised value of a Type II generalised terminant withβ=6,p=1 andq=1/5 using the various MB-regularised forms that apply over the principal branch forz. Because there is no known special function equivalent for this asymptotic series, the results from this study serve as a test-bed for the results in the following chapter. Nevertheless, it is found that the regularised values obtained from the two MB-regularised forms for each of the six common regions of the overlapping domains of convergence equal one another for both small and large values of|z|.


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