This paper considers a generalization of the classical Euler formula to the case of a fourth-order hypercomplex imaginary unit whose square is equal to the hyperbolic unit. This extension unites trigonometric and hyperbolic functions into a single algebraic structure that naturally describes processes in four-dimensional space. It is shown that the exponential with this unit can be decomposed into trigonometric and hyperbolic parts, and the resulting identities and idempotent representation allow us to establish deep connections between ordinary and hyperbolic analysis.

Earlier we obtained an expression for the hyperbolic analog of Euler's formula for 4D space: \[\tag{1} \exp(\ia) = S_0 + \is^1 S_1 + \is^2 S_2 + \is^3 S_3 \\ S_0 = \frac12 (\ch\a + \cos\a) \\ S_1 = \frac12 (\sh\a + \sin\a) \\ S_2 = \frac12 (\ch\a - \cos\a) \\ S_3 = \frac12 (\sh\a - \sin\a) \\ \it = \sqrt{\mathstrut \i}, \quad \i^2 = +1 \] Here: \(\ch, \sh\) are the hyperbolic cosine and sine [1], \(\i\) is the hyperbolic number [2], \(\it\) is the fourth-order hypercomplex imaginary unit.

We will obtain useful identities for partial sums \(S_0 \ldots S_3\), moving from simple expressions to more complex ones. \[\tag{2} \cos\a = S_0 - S_2, \quad \sin\a = S_1 - S_3 \\ \ch\a = S_0 + S_2, \quad \sh\a = S_1 + S_3 \] Sums and differences of squares of pairs
\[\tag{3} (S_0 + S_2)^2 - (S_1 + S_3)^2 = 1 \\ (S_0 - S_2)^2 + (S_1 - S_3)^2 = 1 \] Basic identities for paired products
\[\tag{4} S_0 S_2 + S_1 S_3 = \frac12 \sh^2 \a \\ S_0 S_2 - S_1 S_3 = \frac12 \sin^2 \a \] Useful Square Identities
\[\tag{5} S_0^2 - S_2^2 = \ch\a \cdot \cos\a \\ S_1^2 - S_3^2 = \sh\a \cdot \sin\a \\ S_1^2 + S_3^2 = 2 S_0 S_2 \\ S_0^2 + S_2^2 - 2 S_1 S_3 = 1 \] Armed with these identities, let's move on to the properties of the Euler-type formula with a fourth-order hypercomplex imaginary unit.

Let's take a closer look at the behavior and internal laws of the hypercomplex exponential \(\exp(\i\a)\), which defines the relationship between trigonometric and hyperbolic functions in four-dimensional space and forms the basis for describing its algebraic and geometric properties.

1. Euler-type formula for \(ι\)
\[\tag{6} \exp(\ia) = S_0 + \it S_1 + \ik S_2 + \is^3 S_3, \quad \i = \is^2, \quad \is^4 = 1 \] 2. Conjugates and parities

From here \[\tag{7} \exp(-\ia) = S_0 - \it S_1 + \ik S_2 - \is^3 S_3 \]

3. Product (additivity of the exponent)
since ι commutes with the real numbers, \[\tag{8} \exp \ia \cdot \exp \ib = \exp \it (\a + \b), \] which in the basis \(\{1, \it, \i, \is^3\}\) yields convolution formulas for \(S_k(\a + \b)\). For example, \[\tag{9} S_0(\a + \b) = S_0(\a) S_0(\b) + S_2(\a) S_2(\b) + S_1(\a) S_3(\b) + S_3(\a) S_1(\b) \\ S_1(\a + \b) = S_0(\a) S_1(\b) + S_1(\a) S_0(\b) + S_2(\a) S_3(\b) + S_3(\a) S_2(\b) \]

4. Connection with "Classical" Exponents
\[\tag{10} \exp \i a = (S_0 + S_2) + \i (S_1 + S_3) \\ \exp i a = (S_0 - S_2) + i (S_1 - S_3) \] where: \(i\) is the classical imaginary unit [3].

First, let's get the basic trigonometric and hyperbolic functions with \(\it\) in the exponent: \[\tag{11} \cos \ia = \frac12 \left[ (\ch\a + \cos\a) - \ik (\ch\a - \cos\a) \right] \\ \sin \ia = \frac{\it}{2} \left[ (\sh\a + \sin\a) - \ik (\sh\a - \sin\a) \right] \\ \ch \ia = \frac12 \left[ (\ch\a + \cos\a) + \ik (\ch\a - \cos\a) \right] \\ \sh \ia = \frac{\it}{2} \left[ (\sh\a + \sin\a) + \ik (\sh\a - \sin\a) \right] \] Now, introducing some simplifications for brevity \[ S_k(\a) = S_k^{\a}, \quad S_k(\b) = S_k^{\b}, \] we derive the products of two angles: \[\tag{12} \cos\ia \, \cos\ib = (S_0^{\a} S_0^{\b} + S_2^{\a} S_2^{\b}) - \ik (S_0^{\a} S_2^{\b} + S_2^{\a} S_0^{\b}) \\ \sin\ia \, \sin\ib = -(S_1^{\a} S_3^{\b} + S_3^{\a} S_1^{\b}) + \ik (S_1^{\a} S_1^{\b} + S_3^{\a} S_3^{\b}) \\ \cos\ia \, \sin\ib = \it (S_0^{\a} S_1^{\b} + S_2^{\a} S_3^{\b}) - \is^3 (S_0^{\a} S_3^{\b} + S_2^{\a} S_1^{\b}) \] The results obtained show that operations with the hypercomplex imaginary unit preserveThey take the familiar structure of trigonometric relations but reveal them in a broader, four-dimensional context. This creates the basis for moving on to the next step—considering the internal connections between different imaginary units and constructing their common idempotent representation.

In formula (10), we already tried to make such a connection. Let's generalize this a bit.

Consider the following formula: \[\tag{13} \is = {1 + \i \over 2} + i {1 - \i \over 2} = {1 + i \over 2} + \ik {1 - i \over 2} \] It is easy to verify that: \(\is^2 = \i\).

Based on this, we introduce an idempotent \[\tag{14} \nu_{\pm} = {1 \pm \i \over 2} \] with the following properties: \[\tag{15} \nu_{\pm}^2 = \nu_{\pm}, \quad \nu_{+} \nu_{-} = 0, \quad \ik \nu_{\pm} = \pm\nu_{\pm} \] Then \[\tag{16} \is = \nu_{+} + i \nu_{-} \] and then \[\tag{17} \exp \ia = \nu_{+} \exp(\a) + \nu_{-} \exp(i \a) \]

The transformation of sines and cosines with \(\it\) in the exponent looks very unusual in idempotent and imaginary unit representations: \[\tag{19} \cos\ia = \nu_{+} \cos\a + \nu_{-} \ch\a \\ \sin\ia = \nu_{+} \sin\a + i \nu_{-} \sh\a \\ \ch\ia = \nu_{+} \ch\a + \nu_{-} \cos\a \\ \sh\ia = \nu_{+} \sh\a + i \nu_{-} \sin\a \] Where does the exponential function, where the exponent simultaneously contains an imaginary unit and a fourth-order hypercomplex unit, follow? \[\tag{20} \exp(i\ia) = \nu_{+} \exp(i\a) + \nu_{-} \exp(-\a) \] Also of interest may be the cosine and sine functions, whose exponent simultaneously contains an imaginary unit and a fourth-order hypercomplex unit: \[\tag{21} \cos(i\ia) = \nu_{+} \cos\a + \nu_{-} \ch\a \\ \sin(i\ia) = \nu_{+} \sin\a + i \nu_{-} \sh\a \] where the values ​​of the corresponding formulas are the same as in (19). In this case, the fourth-order hypercomplex unit absorbs the imaginary unit.

The products of sines and cosines for two angles look similar: \[\tag{22} \cos\ia \, \cos\ib = \nu_{+} \cos\a\, \cos\b + \nu_{-}\, \ch\a\kern3pt \ch\b \\ \sin\ia \, \sin\ib = \nu_{+} \sin\a\, \sin\b - \nu_{-}\, \sh\a\kern3pt \sh\b \\ \cos\ia \, \sin\ib = \nu_{+} \cos\a\, \sin\b + i \nu_{-}\, \ch\a\kern3pt \sh\b \] The sum of two angles also looks unusual: \[\tag{23} \cos \is (\a \pm \b) = \nu_{+} \cos (\a \pm \b) + \nu_{-}\, \ch\! (\a \pm \b) \\ \sin \is (\a \pm \b) = \nu_{+} \sin (\a \pm \b) + i \nu_{-}\, \sh\! (\a \pm \b) \\ \ch \is (\a \pm \b) = \nu_{+}\, \ch\! (\a \pm \b) + \nu_{-} \cos (\a \pm \b) \\ \sh \is (\a \pm \b) = \nu_{+}\, \sh\! (\a \pm \b) + i \nu_{-} \sin (\a \pm \b) \] Thus, the introduced fourth-order hypercomplex imaginary unit \(\is\), satisfying \(\is^2 = \i, \i^2 = 1\), allows us to generalize the classical Euler formula. The resulting expressions connect hyperbolic and trigonometric functions in a single algebraic structure, providing a compact description of four-dimensional rotations and dilations.