This monography is intended to be an introduction to the field of quaternionic analysis in a more accessible way. I believe in free knowledge, so I decided to make the book available here for everyone and completely for free in english and italian. For readers who prefer a physical copy, however, I decided to also put a version for sale on amazon.
The book starts with a treatise of the algebraic properties of the quaternions, their algebraic construction (via the Cayley-Dickson construction), the properties of quaternionic polynomials and the study of the topological structures of their zero sets; this is done in the second chapter.
The first chapter, in fact, is a recap of all of the facts from differential geometry, general topology and algebraic topology that will be used throughout the text, in order to make the monography more accessible. This aim was however quite difficult, since all the areas of mathematics cited are by themselves huge; despite this I tried my best to give the minimum required of these subjects in a very short amount of pages in order to make the rest of the text intelligible even to someone who doesn't have a solid foundation in higher mathematics.
The third chapter covers the geometric properties of quaternions; it proves, by constructing the double covering of SO(3) by SU(2), that unit quaternions can be used to represent 3d rotations, and it then goes over the famous phenomenon of "gimbal lock" and shows how quaternions can be a solution to the latter. Finally, it is also shown how quaternions can be employed for spherical interpolation of rotations.
The fourth chapter is devoted to the topology of quaternions, quaternion sequences, quaternionic continuos functions, quaternionic series and the compactification of quaternions. Furthermore, a proof of the Eilenberg-Niven theorem is also provided.
In chapter five we go through all the generalizations of elementary functions (sine,cosine,exponential,logarithm,sinh,cosh,tanh,tan exc.) to the skew-field of quaternions, and we prove some results about them. Below I included a graphic visualization of the quaternionic hyperbolic sine function.
Chapter six introduces the notion of quaternionic left and right regularity, in the sense of Fueter, and then proceeds to prove a quaternionic analogue of the Cauchy-Riemann equations (in this case called Cauchy-Riemann-Fueter equations), Cauchy's theorem, Cauchy's integral formula, Morera's theorem, Liouville's theorem and Taylor and Laurent series expansions.
Chapter seven goes through 3 alternative ways of developing analysis on quaternions:
- Analysis on Banach spaces.
- Analysis on Banach modules.
- The theory of slice regular quaternionic functions.