Recovering holomorphic functions from their real or imaginary parts without the Cauchy-Riemann equations

Students of elementary complex analysis usually begin by seeing the derivation of the Cauchy--Riemann equations. A topic of interest to both the development of the theory and its applications is the reconstruction of a holomorphic function from its real part, or the extraction of the imaginary part from the real part, or vice versa. Usually this takes place by solving the partial differential system embodied by the Cauchy-Riemann equations. Here I show in general how this may be accomplished by purely algebraic means. Several examples are given, for functions with increasing levels of complexity. The development of these ideas within the Mathematica software system is also presented. This approach could easily serve as an alternative in the early development of complex variable theory

Microanalysis of dissolved iron and phosphate in pore waters of hypersaline sediment

Diurnal fluctuations of reduced iron concentrations, expected to occur in reduced sediments in the photic zone, were studied. Iron concentration was compared to O2-H2S, a microcanalysis of sulfate reduction was performed, as well as an examination of diurnal concentration of dissolved phosphate and changes in interstitial CO2. The iron profiles suggest a strong correlation between iron remobilization and processes occurring in the light. Phosphate profiles suggest the removal of phosphate is strongly correlated with precipitation of oxidized iron in the upper 2 mm to 5 mm of the sediments. Pore water CO2 concentrations and carbon isotope ratios are presented. These data are from the analyses of minisediment cores collected from the 42 per mil salt pond and incubated in the laboratory under light and dark conditions

Modelling bonds and credit default swaps using a structural model with contagion

This paper develops a two-dimensional structural framework for valuing credit default swaps and corporate bonds in the presence of default contagion. Modelling the values of related firms as correlated geometric Brownian motions with exponential default barriers, analytical formulae are obtained for both credit default swap spreads and corporate bond yields. The credit dependence structure is influenced by both a longer-term correlation structure as well as by the possibility of default contagion. In this way, the model is able to generate a diverse range of shapes for the term structure of credit spreads using realistic values for input parameters