This paper examines the relationship between the complex unit \(i\), satisfying \(i^2=-1\), and the hyperbolic unit \(\j\), satisfying \(\j^2=+1\). It is shown that, under natural requirements imposed on a mapping from the complex unit circle into the algebra of the hyperbolic unit, its linear form is uniquely determined.

On the basis of this mapping, an expression for the fractional power of the hyperbolic unit is derived: \[ \j^\alpha= \frac12 \left[ \left(1+e^{i\pi\alpha}\right) + \j\left(1-e^{i\pi\alpha}\right) \right], \] and a formula for its complex logarithm is obtained: \[ \ln \j=i\pi(2k+1),\qquad k\in\mathbb Z. \] The main consequence is the introduction of the complex phase of the hyperbolic unit: \[ \phi(\j)=-i\ln\j, \] which, for the principal branch, takes the value \[ \phi(\j)=\pi. \]

Thus, the number \(\pi\) appears not only as the phase corresponding to \(-1\) in ordinary complex algebra, but also as a phase characteristic of the hyperbolic unit within the constructed mapping.

The complex unit \(i\) and the hyperbolic unit \(\j\) are usually treated as distinct algebraic objects. The former defines the complex plane and is associated with rotation, since \[ i^2=-1. \] The latter defines the hyperbolic algebra and satisfies \[ \j^2=+1. \]

For the complex unit, Euler's exponential formula plays a fundamental role: \[ e^{i\theta}=\cos\theta+i\sin\theta, \] describing motion along the unit circle. For the hyperbolic unit, another expression is usually used: \[ e^{\j x}=\cosh x+\j\sinh x. \] However, this form does not establish a direct connection between \(i\) and \(\j\).

In this work, a different approach is considered. Instead of starting from the hyperbolic exponential, we formulate the problem of constructing a mapping from the complex unit circle into the algebra of the hyperbolic unit. Such a mapping must reproduce the basic integer powers of \(\j\) and allow its fractional powers to be defined naturally.

The known powers of the hyperbolic unit are \[ \j^0=1,\qquad \j^1=\j,\qquad \j^2=1. \] Therefore, the desired mapping must send the initial point of the complex unit circle to unity, and the point with phase \(\pi\) to the hyperbolic unit.

We seek a mapping from the complex unit circle \[ e^{i\theta} \] into the algebra generated by the basis \(\{1,\j\}\). We write it in the form \[ \Phi:\;e^{i\theta}\longrightarrow A(\theta)+\j B(\theta), \] where \(A(\theta)\) and \(B(\theta)\) are coefficients depending on the complex phase.

We require the mapping to satisfy two basic conditions: \[ \Phi(1)=1, \] \[ \Phi(-1)=\j. \] The first condition corresponds to \(\j^0=1\). The second condition corresponds to \(\j^1=\j\), since the point \(-1\) on the complex unit circle corresponds to the phase \(\theta=\pi\).

Since the elements \(1\) and \(\j\) form the basis of the algebra under consideration, it is natural to represent the image of the complex phase as a linear combination of these basis elements. The minimal assumption is that the coefficients \(A\) and \(B\) depend linearly on the complex exponential \(e^{i\theta}\).

Consider a mapping of the form \[ \Phi(e^{i\theta})=A+\j B, \] where the coefficients \(A\) and \(B\) depend linearly on the complex exponential: \[ A=a+b e^{i\theta}, \] \[ B=c+d e^{i\theta}. \] Then, among mappings of this form, there exists a unique mapping satisfying \[ \Phi(1)=1, \] \[ \Phi(-1)=\j. \]

Substituting the linear expressions for \(A\) and \(B\) into the general representation of the mapping gives \[ \Phi(e^{i\theta})= a+b e^{i\theta} + \j\left(c+d e^{i\theta}\right). \] We first use the condition \[ \Phi(1)=1. \] For \(e^{i\theta}=1\), we obtain \[ A(1)=a+b=1, \] \[ B(1)=c+d=0. \]

We now use the second condition: \[ \Phi(-1)=\j. \] For \(e^{i\theta}=-1\), we obtain \[ A(-1)=a-b=0, \] \[ B(-1)=c-d=1. \] Thus, the problem reduces to solving two independent systems of linear equations.

For the coefficient \(A\), we have the system \[ a+b=1, \] \[ a-b=0. \] Adding these equations gives \[ 2a=1, \] hence \[ a=\frac12. \] Therefore, \[ b=\frac12. \]

For the coefficient \(B\), we have the system \[ c+d=0, \] \[ c-d=1. \] Adding these equations gives \[ 2c=1, \] hence \[ c=\frac12. \] Therefore, \[ d=-\frac12. \]

Thus, \[ A=\frac12\left(1+e^{i\theta}\right), \] \[ B=\frac12\left(1-e^{i\theta}\right). \] Consequently, the required mapping is \[ \boxed{ \Phi(e^{i\theta})= \frac12 \left[ \left(1+e^{i\theta}\right) + \j\left(1-e^{i\theta}\right) \right]. } \]

Since the system of linear equations for the coefficients \(a\), \(b\), \(c\), and \(d\) has a unique solution, the mapping obtained above is the unique linear mapping of the specified form satisfying \(\Phi(1)=1\) and \(\Phi(-1)=\j\). The theorem is proved.

Set \[ \theta=\pi\alpha. \] Then the complex phase \(\theta\) is expressed through the exponent \(\alpha\), and the constructed mapping allows us to define fractional powers of the hyperbolic unit: \[ \boxed{ \j^\alpha= \frac12 \left[ \left(1+e^{i\pi\alpha}\right) + \j\left(1-e^{i\pi\alpha}\right) \right]. } \]

Let us verify the basic values. For \(\alpha=0\): \[ \j^0= \frac12 \left[ \left(1+1\right) + \j\left(1-1\right) \right] =1. \] For \(\alpha=1\): \[ \j^1= \frac12 \left[ \left(1+e^{i\pi}\right) + \j\left(1-e^{i\pi}\right) \right]. \] Since \[ e^{i\pi}=-1, \] we obtain \[ \j^1= \frac12 \left[ \left(1-1\right) + \j\left(1+1\right) \right] =\j. \]

For \(\alpha=2\): \[ \j^2= \frac12 \left[ \left(1+e^{2i\pi}\right) + \j\left(1-e^{2i\pi}\right) \right]. \] Since \[ e^{2i\pi}=1, \] we obtain \[ \j^2= \frac12 \left[ \left(1+1\right) + \j\left(1-1\right) \right] =1. \] Therefore, the obtained formula is fully consistent with the condition \(\j^2=1\).

Since the complex exponential is periodic, \[ e^{i\pi(\alpha+2)}= e^{i\pi\alpha}e^{2i\pi} = e^{i\pi\alpha}, \] the fractional powers of the hyperbolic unit satisfy \[ \j^{\alpha+2}=\j^\alpha. \] Thus, the periodicity of the powers of \(\j\) follows directly from the periodicity of the complex exponential.

This property shows that the constructed fractional power preserves the basic cyclic structure of the hyperbolic unit: \[ 1\longrightarrow \j\longrightarrow 1. \] The transition from \(1\) to \(\j\) corresponds to a change of the complex phase by \(\pi\), while the full return to \(1\) corresponds to a phase change of \(2\pi\).

For intermediate values of the parameter \(\alpha\), the mapping takes us beyond the basis \(\{1,\j\}\) and naturally leads to the extended algebra \[ \{1,i,\j,i\j\}. \] For example, for \[ \alpha=\frac12 \] we have \[ e^{i\pi\alpha}=e^{i\pi/2}=i. \]

Then \[ \j^{1/2} = \frac12 \left[ (1+i)+\j(1-i) \right], \] or \[ \boxed{ \j^{1/2} = \frac12 \left(1+i+\j-i\j\right). } \]

This shows that fractional powers of the hyperbolic unit naturally connect the complex unit \(i\), the hyperbolic unit \(\j\), and the product \(i\j\). Therefore, the constructed mapping can be regarded as a bridge between complex and hyperbolic structures.

We now define the logarithm of the hyperbolic unit as a quantity satisfying the relation \[ \j^\alpha=e^{\alpha\ln \j}. \] Within the constructed framework, the dependence on the exponent \(\alpha\) is determined by the phase factor \[ e^{i\pi\alpha}. \] Therefore, the logarithm \(\ln\j\) must reproduce a phase factor of the form \[ e^{\alpha\ln\j}=e^{i\pi\alpha}. \]

The principal branch is \[ \boxed{ \ln \j=i\pi. } \] It is important to emphasize that this logarithmic value is not introduced in advance; it arises as a consequence of the constructed mapping from the complex unit circle into the algebra of the hyperbolic unit.

The result can also be expressed in terms of the mapping \(\Phi\). Since \[ e^{i\pi}=-1, \] and, by construction, \[ \Phi(-1)=\j, \] it follows that \[ \Phi(e^{i\pi})=\j. \] This is precisely what leads to the principal branch \[ \ln\j=i\pi. \]

In complex analysis, a quantity associated with the expression \[ -i\ln z \] may be regarded as a phase characteristic of a complex number. In the present construction, an analogous quantity can be introduced for the hyperbolic unit.

Define the complex phase of the hyperbolic unit by \[ \phi(\j)=-i\ln\j. \] Using the expression obtained for the logarithm, \[ \ln \j=i\pi(2k+1), \] we get \[ \phi(\j) = -i\ln\j = -i\cdot i\pi(2k+1). \] Since \[ -i\cdot i=1, \] it follows that \[ \boxed{ \phi(\j)=(2k+1)\pi, \qquad k\in\mathbb Z. } \]

For the principal branch, we obtain the particularly simple expression \[ \boxed{ \phi(\j)=-i\ln\j=\pi. } \] In other words, within the constructed mapping, the hyperbolic unit \(\j\) corresponds to the complex phase \(\pi\).

This value is not arbitrary. It appears because the point on the complex unit circle with phase \(\pi\) is equal to \(-1\), and the constructed mapping sends this point to the hyperbolic unit: \[ \Phi(e^{i\pi})=\j. \] Therefore, \(\pi\) acts as the phase characteristic of the hyperbolic unit.

In ordinary complex algebra, Euler's formula gives the well-known relation \[ e^{i\pi}=-1. \] It shows that the phase \(\pi\) on the complex unit circle corresponds to the number \(-1\).

In the construction presented here, the same complex phase \(\pi\), through the mapping \(\Phi\), corresponds instead to the hyperbolic unit: \[ \Phi(e^{i\pi})=\j. \] Thus, the hyperbolic unit acquires its own complex-phase interpretation.

Therefore, the formula \[ -i\ln\j=\pi \] for the principal branch plays the role of a phase analogue of the well-known Euler relation. It does not assert that \(\j\) is equal to \(-1\) in ordinary complex algebra; rather, it shows that \(\j\) corresponds to the phase \(\pi\) within the constructed mapping of the complex unit circle.

A mapping from the complex unit circle into the algebra of the hyperbolic unit has been constructed. It has been shown that, among linear mappings of the form \[ \Phi(e^{i\theta})= a+b e^{i\theta} + \j\left(c+d e^{i\theta}\right), \] there exists a unique mapping satisfying \[ \Phi(1)=1, \] \[ \Phi(-1)=\j. \] It is given by \[ \boxed{ \Phi(e^{i\theta})= \frac12 \left[ \left(1+e^{i\theta}\right) + \j\left(1-e^{i\theta}\right) \right]. } \]

On the basis of this mapping, the formula for the fractional power of the hyperbolic unit has been obtained: \[ \boxed{ \j^\alpha= \frac12 \left[ \left(1+e^{i\pi\alpha}\right) + \j\left(1-e^{i\pi\alpha}\right) \right]. } \] This formula is not an arbitrary assumption; it follows from the problem of constructing a linear mapping consistent with the basic properties of the hyperbolic unit.

The logarithm of the hyperbolic unit has also been obtained: \[ \boxed{ \ln \j=i\pi(2k+1),\qquad k\in\mathbb Z. } \] For the principal branch: \[ \boxed{ \ln \j=i\pi. } \]

The main result of the work is the introduction of the complex phase of the hyperbolic unit: \[ \phi(\j)=-i\ln\j. \] For all branches: \[ \boxed{ \phi(\j)=(2k+1)\pi, \qquad k\in\mathbb Z. } \] For the principal branch: \[ \boxed{ \phi(\j)=\pi. } \]

Thus, a direct connection is established between the complex unit \(i\), the hyperbolic unit \(\j\), the extended algebra \(\{1,i,\j,i\j\}\), and the phase of the complex exponential. The number \(\pi\) receives an additional interpretation as the complex phase of the hyperbolic unit within the constructed mapping.