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This book describes and illustrates the application of several
asymptotic methods that have proved useful in the authors' research
in electromagnetics and antennas. We first define asymptotic
approximations and expansions and explain these concepts in detail.
We then develop certain prerequisites from complex analysis such as
power series, multivalued functions (including the concepts of
branch points and branch cuts), and the all-important gamma
function. Of particular importance is the idea of analytic
continuation (of functions of a single complex variable); our
discussions here include some recent, direct applications to
antennas and computational electromagnetics. Then, specific methods
are discussed. These include integration by parts and the
Riemann-Lebesgue lemma, the use of contour integration in
conjunction with other methods, techniques related to Laplace's
method and Watson's lemma, the asymptotic behavior of certain
Fourier sine and cosine transforms, and the Poisson summation
formula (including its version for finite sums). Often
underutilized in the literature are asymptotic techniques based on
the Mellin transform; our treatment of this subject complements the
techniques presented in our recent Synthesis Lecture on the exact
(not asymptotic) evaluation of integrals.
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