Complex Multiplicative Calculus and Mixed Problems

Sammanfattning: In this thesis we consider multiplicative integrals and derivatives on functions on the positive numbers, the logarithmic Riemann surface and the punctured complex numbers. Additive structures on the real and complex numbers are related to their multiplicative counterparts on the positive numbers and the logarithmic Riemann surface by defining exponential transition functors. We use a lift-projection method to transition multiplicative structures on the logarithmic Riemann Surface to their counterparts on the punctured complex plane. The process introduces potentially multivalued behaviour, which is the case for multiplicative integrals on functions with a range in the punctured complex plane. Some mixed problems, involving both additive and multiplicative structures, are also discussed. E.g. we consider the mixed differential equation y’= y', whose solution involves the Lambert W function. We extend the inequality of arithmetic and geometric means (AM-GM inequality) to the setting of non-negative random variables. A matrix version of the AM-GM inequality is also extended, and tweaked to an integral version which leads to a generalization of Hölder’s inequality.