Complex and Harmonic Analysis
Complex analysis is one of the classical branches of mathematics, but still a vigorous field in rapid development. It is useful in many other mathematical disciplines, like functional analysis, differential equations and number theory.
In addition, it is a powerful tool in numerous areas of engineering (for example hydrodynamics, signal analysis and control theory) and theoretical physics. These applications again lead to exciting new problems in complex analysis.
The fundamental idea of harmonic analysis is to develop a signal into a collection of pure waves (harmonics), and then use this development to analyze the signal, including its compression and transmission. This idea of dividing an object into simple basic parts and analyzing this division has proved to be very efficient within a wide range of mathematical disciplines and fields of engineering. It is the collection of ideas and methods connected to this approach that constitute the core of harmonic analysis. Many of the tools of harmonic analysis are related to ideas in complex analysis.