Part I: Single Variables: Review and extensions. CR equations and analyticity; Cauchy-Goursat theorem and Cauchy integral formula, Louiville’s theorem; Morera’s theorem; maximum modulus principle; Laurent series and singularities; Riemann extension theorem; residues; Schwartz’s lemma; open mapping theorem; analytic continuation and the dilogarithm; linear fractional transformations; spaces of a...

In this paper, we determine periodicity of quantum walks defined by mixed paths and cycles. By the spectral mapping theorem walks, consideration is reduced to eigenvalue analysis $\eta$-Hermitian adjacency matrices. First, investigate coefficients characteristic polynomials We show that trees their underlying graphs are same. also define $n+1$ types cycles every cycle switching equivalent one t...

We will develop some of the basic concepts of complex function theory and prove a number of useful results concerning holomorphic functions. We will focus on derivatives, zeros, and sequences of holomorphic functions. This will lead to a brief discussion of the significance of biholomorphic mappings and allow us to prove the Riemann mapping theorem.