Applied and Computational Complex Analysis Vol 2 - P. Henrici
In the present Volume II of our three-volume work we continue to discuss
algorithmic techniques that can be used to construct either exact or approxi-
mate solutions to problems in complex analysis. A focal point for these
applications is the evaluation and manipulation of solutions of analytic
differential equations. Successive chapters deal with the representation of
solutions by (convergent or divergent) series expansions, with the method of
integral transforms, with asymptotic analysis, and with the representation
of special solutions by continued fractions. The gamma function is dealt with
in the opening chapter in the context of product expansions of analytic
functions.
Together with its companions, this volume provides a fair amount of
information on some of the more important special functions of mathemati-
cal physics. However, our treatment of these functions is unconventional in
its organization. Whereas the conventional treatment proceeds function by
function, giving to each function its due share of series and integral represen-
tations, and of asymptotic analysis, our treatment proceeds by general
methods and problems rather than by individual functions. Special results
thus appear mainly as applications of general principles. The same
methodology will be followed in Volume III; for instance, addition theorems
will be considered in the context of partial differential equations.
Although I hope that my program has enabled me to illuminate the basic
properties of special functions such as the gamma function, the
hypergeometric function, the confluent hypergeometric function, and the
Bessel functions, it must be pointed out that a full in-depth treatment of any
class of special functions was neither intended nor possible. For more
detailed information the reader should turn either to specialized treatises or
to the monumental Bateman manuscript project (Erd61yi [1953], [1955]),
which provides an essentially complete collection of results known up to the
early 1950s.
In the present Volume II of our three-volume work we continue to discuss
algorithmic techniques that can be used to construct either exact or approxi-
mate solutions to problems in complex analysis. A focal point for these
applications is the evaluation and manipulation of solutions of analytic
differential equations. Successive chapters deal with the representation of
solutions by (convergent or divergent) series expansions, with the method of
integral transforms, with asymptotic analysis, and with the representation
of special solutions by continued fractions. The gamma function is dealt with
in the opening chapter in the context of product expansions of analytic
functions.
Together with its companions, this volume provides a fair amount of
information on some of the more important special functions of mathemati-
cal physics. However, our treatment of these functions is unconventional in
its organization. Whereas the conventional treatment proceeds function by
function, giving to each function its due share of series and integral represen-
tations, and of asymptotic analysis, our treatment proceeds by general
methods and problems rather than by individual functions. Special results
thus appear mainly as applications of general principles. The same
methodology will be followed in Volume III; for instance, addition theorems
will be considered in the context of partial differential equations.
Although I hope that my program has enabled me to illuminate the basic
properties of special functions such as the gamma function, the
hypergeometric function, the confluent hypergeometric function, and the
Bessel functions, it must be pointed out that a full in-depth treatment of any
class of special functions was neither intended nor possible. For more
detailed information the reader should turn either to specialized treatises or
to the monumental Bateman manuscript project (Erd61yi [1953], [1955]),
which provides an essentially complete collection of results known up to the
early 1950s.
