By Klaus Gürlebeck
Complex research these days has higher-dimensional analoga: the algebra of complicated numbers is changed then via the non-commutative algebra of genuine quaternions or via Clifford algebras. over the past 30 years the so-called quaternionic and Clifford or hypercomplex research effectively built to a strong idea with many purposes in research, engineering and mathematical physics. This textbook introduces either to classical and higher-dimensional effects in keeping with a uniform thought of holomorphy. historic comments, plenty of examples, figures and workouts accompany every one bankruptcy.
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Extra info for Holomorphic Functions in the Plane and n-dimensional Space
48 Chapter I. 53 (Spherical cosine theorem). We have cos β = cos γ cos α + sin γ sin α cos β . Proof. Let a, b, c be vectors with |a| = |b| = |c| = 1. The Lagrange identity delivers a·b b·b (a × b) · (b × c) = a·c b·c = (a · b)(b · c) − a · c = cos γ cos α − cos β . For the left side we obtain (a × b) · (b × c) = sin γ sin α cos β = − sin γ sin α cos β , from which the desired relation follows. 54 (Spherical sine-cosine theorem). With the already introduced notation we have the following relationship: sin α cos γ = cos γ sin β − cos β sin γ cos α .
Conversely each rotation in R3 can be represented by an automorphism of the form ρy = yxy −1 with y ∈ H. 23. 2. The rotation axis of consecutive rotations can then be calculated in an elegant way, see also . Proof. Part a) was proven before the formulation of the proposition, we have only to show part b). Since in fact we have a rotation about the axis ω, we can best proceed by decomposing the vectors x and x in the components parallel to ω and those perpendicular to it. We can easily convince ourselves that this decomposition can be written as x =: z + (ω · x)ω, x =: z + (ω · x )ω.
For x · y = 0, x and y are orthogonal to each other. 8. Let x, y, z be arbitrary elements in H. Then (i) Sc (xyz) = Sc(yzx) = Sc(zxy), (ii) Sc(xyz) = x · (yz) is a real number, which is also called a scalar mixed (or triple) product of the quaternions x, y, z (in this order). Proof. 1). 9. So far we have worked with R, C and H, sets that have both a vector space structure and a ﬁeld structure. They are called algebras. Since all elements diﬀerent from zero have a multiplicative inverse, we speak of division algebras.
Holomorphic Functions in the Plane and n-dimensional Space by Klaus Gürlebeck