By Lokenath Debnath-
Foreign reprint variation. third variation. Paperback. released in China. Contents in ENGLISH and absolutely related as US version. (Follower dealers, resembling "gofrl" or "WhiteCoffee", may perhaps promote you different reasonable variation books)
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1) 32 Chapter 4. Trigonometry in the Minkowski Plane is invariant under axes rotation. In a similar way we find two more invariants related to any pair of vectors. 2. The real and imaginary parts of the product v2 v¯1 are invariant under axes rotations and these two invariants allow us an operative definition of trigonometric functions by means of the components of the vectors: cos(φ2 − φ1 ) = x1 x2 + y1 y2 ; ρ1 ρ2 sin(φ2 − φ1 ) = x1 y2 − x2 y1 . 2) Proof. 3) and let us represent the two vectors in polar coordinates v1 ≡ ρ1 exp[i φ1 ], v2 ≡ ρ2 exp[i φ2 ].
We begin by recalling the importance of linear transformations as related to important geometries. An arbitrary linear transformation can be written as y γ = cγβ xβ , with the condition cγβ = 0; therefore it depends on N 2 parameters. By identifying y γ and xβ as vector components, we can write, following the notation of linear algebra, ⎞⎛ 1 ⎞ ⎛ 1 ⎞ ⎛ 1 c1 · · · c1N x y ⎟ ⎜ ⎜ .. ⎟ ⎜ .. . .. ⎠ ⎝ ... 1) ⎠. ⎝ . ⎠=⎝ . yN cN 1 ··· cN N xN These transformations are known as homographies and are generally non-commutative.
Now, we consider this geometry as the simplest one associated with decomposable systems of numbers. The multiplicative group of hyperbolic numbers, expressed in vector-matrix form, is given by y1 y2 = A11 A21 A12 A22 x1 x2 with A11 = A22 , A12 = A21 . 6) The unimodularity condition requires that (A11 )2 − (A21 )2 = 1 and this allows introducing a hyperbolic angle θ so that A11 = cosh θ, A21 = sinh θ. This position is equivalent to writing, in complex analysis, the constant of the multiplicative group in its polar form.