2025-09-21
Absorption and transfer of the hyperbolic unit by some formulas
The hyperbolic unit, like the imaginary unit, has its own characteristics and rules for manipulation. While the imaginary unit is familiar to everyone from Euler's formula and the properties of trigonometric functions, the hyperbolic unit remains less studied in this context. However, upon closer examination, it turns out that it can naturally be incorporated into familiar formulas, changing their appearance while maintaining their internal harmony. In this paper, we will examine how the hyperbolic unit is absorbed or removed from certain expansions and functions, and how this behavior resembles that of the imaginary unit. This approach allows us to clearly see the hidden connection between trigonometric and hyperbolic expressions and expand our understanding of classical formulas.
Extension of Euler's Formula with a Hyperbolic Unit
We introduce the following hypercomplex unit \[\tag{1} \ii = \i \cdot i \] where: \(i\) is the imaginary unit, whose square is minus one [1], \(\i\) is the hyperbolic unit, whose square is plus one. Then we can find powers of such a unit, according to the rules of hyperbolic and imaginary numbers: \[\tag{2} \ii^2 = -1, \quad \ii^3 = -\ii, \quad \ii^4 = 1 \] These values fully correspond to similar operations with the imaginary unit [1]: \[\tag{3} i^2 = -1, \quad i^3 = -i, \quad i^4 = 1 \] Then, when expanding the exponential in a Maclaurin series [2], by substituting \(\ii\), we obtain the same values of the series terms as by substituting \(i\). Therefore, we can extend Euler's formula [3] with a hyperbolic unit as follows:
\[\tag{4} \exp(\ii \a) = \cos\a + \ii \sin\a, \quad \ii = \i \cdot i \] This is a clear example of the absorption of the hyperbolic unit by the imaginary unit. However, it is not always absorbed by \(i\), but only when the series is expanded in powers of integers. If the powers of such a series are fractional numbers, then the reverse process occurs—the imaginary unit is absorbed. Examples of such series are given here, but we'll discuss them another time, since we're only considering scalar functions now.
Let's now look at what Euler's formula would look like if we left only the hyperbolic unit in its exponent. To do this, let us recall the Maclaurin series expansions of the hyperbolic cosine and sine [4]: \[\tag{5} \ch x = \sumn0 {x^{2n} \over (2n)!} \\ \sh x = \sumn0 {x^{2n+1} \over (2n+1)!} \] and substitute into them \[ x = \ik\a \] where: \(\i\) is the hyperbolic unit, the square of which is equal to plus one.
Considering the rules for working with such units \[\tag{6} \i^n = \left\{\begin{matrix} 1 & \text{if} & n = 0,2,4,\ldots \\ \i & \text{if} & n = 1,3,5,\ldots \end{matrix}\right.\] From series (5), we obtain formulas for the absorption and transfer of the hyperbolic unit by the hyperbolic cosine and sine:
\[\tag{7} \ch\!(\ik\a) = \ch \a \\ \sh\!(\ik\a) = \ik\sh\a \]
It remains to recall [4] that \[\tag{8} \exp x = \ch x + \sh x \] from which we obtain Euler's formula with a hyperbolic unit:
\[\tag{9} \exp(\ik\a) = \ch\a + \ik\sh\a, \quad \i^2 = +1 \] Applying this algorithm, we proceed in the same way with other functions.
Absorption and Transfer of the Hyperbolic Unit by Some Functions
Let's start with the cosine and sine functions. Their expansion in a Maclaurin series is as follows [2]: \[\tag{10} \cos x = \sumn0 (-1)^n {x^{2n} \over (2n)!} \\ \sin x = \sumn0 (-1)^n {x^{2n+1} \over (2n+1)!} \] Substituting into them \[ x = \ik\a \] and using rule (6), we obtain formulas for the absorption and transfer of the hyperbolic unit by the trigonometric cosine and sine:
\[\tag{11} \cos(\ik\a) = \cos \a \\ \sin(\ik\a) = \ik\sin\a \] Compare these expressions with formulas (7).
In the same way, we can obtain the absorption and translation of the hyperbolic unit by some other functions: \[\tag{12} \tan(\ik\a) = \i\tan\a \\ \th\!(\ik\a) = \i\,\th\a \\ \ln(1 + \ik\a) = \i\ln(1 + \a) \] The general pattern is obvious: if, when expanding a function in a Maclaurin series, all powers of \(x\) are even, then the hyperbolic unit is absorbed by this formula; if only odd powers, then it is carried outside the formula, multiplying by it.