by Valery Serov; Markus Harju
The first two parts of this book focus
on developing standard analysis concepts in the extended complex plane.
We cover differentiation and integration of functions of one complex
variable. Famous Cauchy formulas are established and applied in the
frame of residue theory. Taylor series is used to investigate analytic
functions, and they are connected to harmonic functions. Laurent series
theory is developed.
The third
part of the book finds applications of the earlier chapter in conformal
mappings and the Laplace transform. Special functions solving ordinary
differential equations are studied extensively, along with their
asymptotic behavior. A highlight of the book is the elliptic function
of Weierstrass and Jacobi. Finally, we present Laplace’s method, which
is applied to find large arguments asymptotic of some special
functions.
The book is filled with
examples, exercises, and problems of varying degrees of difficulty.
This makes it useful to all students in mathematics, physics, and
related fields.
