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## Über dieses Buch

Real quaternion analysis is a multi-faceted subject. Created to describe phenomena in special relativity, electrodynamics, spin etc., it has developed into a body of material that interacts with many branches of mathematics, such as complex analysis, harmonic analysis, differential geometry, and differential equations. It is also a ubiquitous factor in the description and elucidation of problems in mathematical physics. In the meantime real quaternion analysis has become a well established branch in mathematics and has been greatly successful in many different directions. This book is based on concrete examples and exercises rather than general theorems, thus making it suitable for an introductory one- or two-semester undergraduate course on some of the major aspects of real quaternion analysis in exercises. Alternatively, it may be used for beginning graduate level courses and as a reference work. With exercises at the end of each chapter and its straightforward writing style the book addresses readers who have no prior knowledge on this subject but have a basic background in graduate mathematics courses, such as real and complex analysis, ordinary differential equations, partial differential equations, and theory of distributions.

## Inhaltsverzeichnis

### 1. An Introduction to Quaternions

Once we start studying quaternionic analysis we take part in a wonderful experience, full of insights. This ideology is shown, for instance, when we start describing the first results and pursuing the subject, while the amazement lingers on through the elegance and smoothness of the results.
João Pedro Morais, Svetlin Georgiev, Wolfgang Sprößig

### 2. Quaternions and Spatial Rotation

The particularly rich theory of rotations does not need advertising. One can think of a rotation as a transformation in the plane or in space that describes the position and orientation of a three-dimensional rigid body around a fixed point. The first ever study of rotations was published by L. Euler in 1776.
João Pedro Morais, Svetlin Georgiev, Wolfgang Sprößig

### 3. Quaternion Sequences

In the present chapter we use the properties of quaternions described in a previous chapter to explore the key notion of a quaternion sequence. Then we will use this analogue in a formula called summation by parts, which is an analogue of integration by parts for sums. Summation by parts is not only a useful auxiliary tool, but even indispensable in many applications, including finding sums of powers of integers and deriving some famous convergence tests for series: the Dirichlet and Abel tests.
João Pedro Morais, Svetlin Georgiev, Wolfgang Sprößig

### 4. Quaternion Series and Infinite Products

An essential feature of the classical theory of power series is that we can manipulate recurrence relations for power series without necessarily worrying about whether the underlying series converge. In case they do converge, we can extract important information about the recurrence relation that may not otherwise be easily obtainable.
João Pedro Morais, Svetlin Georgiev, Wolfgang Sprößig

### 5. Exponents and Logarithms

The real exponential and logarithmic functions play an important role in advanced mathematics, including applications to calculus, differential equations, and complex analysis. In this chapter we use the properties of quaternions described in the previous chapters to define and study the quaternionic analogues of these functions.
João Pedro Morais, Svetlin Georgiev, Wolfgang Sprößig

### 6. Trigonometric Functions

In this chapter we define quaternion trigonometric functions. Analogously to the quaternion functions e p and $$\ln (p)$$, these functions will agree with their counterparts for real and complex input. In addition, we will show that the quaternion trigonometric functions satisfy many of the same identities the real and complex trigonometric functions do.
João Pedro Morais, Svetlin Georgiev, Wolfgang Sprößig

### 7. Hyperbolic Functions

After bringing together various results mentioned before, in this chapter we introduce the quaternion hyperbolic functions, whose study will require us to master a new situation. Since the quaternion exponential function agrees with the real and complex exponential function of real and complex arguments, it follows that the quaternion hyperbolic functions also agree with their usual counterparts for real and complex input. This allows us to discuss some important hyperbolic identities and the existence of infinitely many zeros of the quaternion sine and cosine hyperbolic functions, and to solve equations involving hyperbolic functions. A remarkable result of the theory exhibits the deep connection between the hyperbolic and trigonometric functions discussed in the previous chapter. We hope that material presented in this part will make this beautiful topic accessible to the reader.
João Pedro Morais, Svetlin Georgiev, Wolfgang Sprößig

### 8. Inverse Hyperbolic and Trigonometric Functions

The main focus of this chapter is to study the inverses of the quaternion trigonometric and hyperbolic functions, and their properties. Since the quaternion trigonometric and hyperbolic functions are defined in terms of the quaternion exponential function e p , it can be shown that their inverses are necessarily multi-valued and can be computed via the quaternion natural logarithm function ln(p). The s facts we shall see here attest the great interest of these functions in mathematics. Proofs of the most known facts are ommited.
João Pedro Morais, Svetlin Georgiev, Wolfgang Sprößig

### 9. Quaternion Matrices

In a brief outline, the next portion of text describes a way of representing quaternion matrices in such a way that quaternionic addition and multiplication correspond to matrix addition, (scalar) matrix multiplication, and matrix transposition. Besides the discussion of the quaternionic analogues to complex matrices, we will also discuss the possibility of decomposing quaternions with respect to well known matrix decompositions, which are related to solving systems of linear equations. Even though it is self-contained, the reader will comprehend the necessity of this notions while pursuing the subject. Our presentation is organized in such a way that the analogies between quaternion, noncommutativity, and quaternion matrix, on one hand, and determinant, rank, and eigenvalues, on the other, are mastered. The relations between these concepts will come into view more clearly through the text, motivating our insight to the problem.
João Pedro Morais, Svetlin Georgiev, Wolfgang Sprößig

### 10. Monomials, Polynomials and Binomials

In this chapter we will be primarily interested in the study of monomials and polynomials within the framework of quaternion analysis. Monomials and their applications to combinatorics and number theory have become increasingly important for the study of a large number of problems that arise in many different contexts, both from a theoretical and a practical perspective. At the same time, the applications of polynomials to classical and numerical analysis, including approximation theory, statistics, combinatorics, number theory, group representations etc., as well as in physics, including quantum mechanics and statistical physics, and in system theory and signal processing have played a key role in this development and continue to do it today. For example, polynomials are often used in the treatment of problems, mainly in mathematical physics, and also in studies related to differential equations, continued fractions, and numerical stability. In advanced mathematics, polynomials are used to construct polynomial rings, a central concept in abstract algebra and algebraic geometry.
João Pedro Morais, Svetlin Georgiev, Wolfgang Sprößig

### 11. Solutions

João Pedro Morais, Svetlin Georgiev, Wolfgang Sprößig

### Backmatter

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