Complex analysis is the study of functions that live in the complex plane, i.e. functions that have complex arguments and complex outputs. In order to study the behavior of such functions we’ll need to first understand the basic objects involved, namely the complex numbers. We’ll begin with some history: When and why were complex numbers invented? Was it the need for a solution of the equation x^2 = -1 that brought the field of complex analysis into being, or were there other reasons?
Analytic Combinatorics is based on formal methods for deriving functional relationships on generating functions and asymptotic analysis treating those functions as functions in the complex plane. This course covers the symbolic method for defining generating functions immediately from combinatorial constructions, then develops methods for directly deriving asymptotic results from those generating functions, using complex asymptotics, singularity analysis, saddle-point asymptotics, and limit laws. The course teaches the precept "if you can specify it, you can analyze it".
The period of the demise of the Kingdom of Judah at the end of the sixth
century B.C.E., the fall of Jerusalem to the Babylonians, the exile of
the elite to Babylon, and the reshaping of the territory of the new province
of Judah, culminating at the end of the century with the first return of
exiles – all have been subjects of intense scrutiny in modern scholarship.
This course takes into account the biblical textual evidence, the results
of archaeological research, and the reports of the Babylonian and Egyptian