Applications of Matching Conformal Mappings in Analytic Functions

Abstract

Conformal maps are the basic tool in complex analysis as they retain local geometry, and allow one to homogenize complicated domains into simple regions. In this paper, an exact analytic and computational approach for matching conformal maps in analytic functions is introduced focused on complex potentials, composition-type transformations, and spectral transfer matrices. The framework provides the Cauchy-Riemann, conformal metric invariance, harmonic-function preservation, and transmission laws for the well-posedness of the boundary-value problems with Laplace and transient heat equations for each subdomain. It then extends the methodology to non-constant physical domains, notably layered desert soil surrounding a half-buried cylindrical pipeline, where Fourier decomposition, fractional Robin surface memory, block-tridiagonal transfer matrices, generalized Joukowski mapping, and pipe-wall flux integrals are merged in a single verifiable solution line. The formulation developed relates analytic-function theory to heat-transfer diagnostics, the temperature field, electrostatic and potential-theory analogues, and engineering interpretation of complex geometries. The investigation demonstrates that the theories of matching conformal maps can transform highly irregular boundary configurations to standard computational domains while maintaining the harmonic structure required for effective analytic and numerical solutions.