We define a hyperholomorphic function with values in split quaternions, provide split hyperholomorphic mappings on ­ and research the properties of split hyperholomorphic functions. A set of quaternions can be represented as where and e ₁ e ₂ e ₃ = −1, which is non-commutative division algebra. A set of split quaternions can be expressed as where and e ₁ e ₂ e ₃ = 1, which is also non-commutative. On the other hand, unlike quaternion algebra, a set of split quaternions contains zero divisors, nilpotent elements and non-trivial idempotents. Because split quaternions are used to express Lorentzian rotations, studies of the geometric and physical appli-cations of split quaternions require solving split quaternionic equations (see [6] , [9] ). Deavours [3] generated regular functions in a quaternion analysis and provided the Cauchy-Fueter integral formulas for regular quaternion functions. Carmody [1 , 2] investigated the properties of hyperbolic quaternions, octonions, and sedenions, and Sangwine and Bihan [10] provided a new polar representation of quaternions that is represented by a pair of complex numbers in the Cayley-Dickson form. We shall denote by and , respectively, the field of complex numbers and the field of real numbers. We [4 , 5] showed that any complex-valued harmonic function f ₁ in a pseudoconvex domain D of has a conjugate function f ₂ in D such that the quaternion-valued function f ₁ + f ₂ j is hyperholomorphic in D and gave a regeneration theorem in a quaternion analysis in view of complex and Clifford analysis method. We define a split hyperholomorphic function with values in split quaternions and examine the properties of split hyperholomorphic functions based on [7] . The split quaternionic field S is a four-dimensional non-commutative -field generated by four base elements e ₀ , e ₁ , e ₂ , and e ₃ with the following non-commutative multiplication rules : The element e ₀ is the identity of S , and e ₁ identifies the imaginary unit in the -field of complex numbers. A split quaternion z is given by where z ₁ = x ₀ + e ₁ x ₁ , z ₂ = x ₂ + e ₁ x ₃ , and are complex numbers in and x_k ( k = 0, 1, 2, 3) are real numbers. The multiplications of two pure split quaternions and is defined as follows: For pure split quaternions , and , the cross product satisfies two rules as follows: The split quaternionic conjugate z *, the multiplicative modulus M ( z ) and the inverse z ⁻¹ of z in S are defined as We let The split quaternion number z of S is where ξ ₀ = x ₀ and Then the split quaternionic conjugate number of z is z * = ξ ₀ − Jξ ₁ , and the multiplicative modulus of z is Let Ω be an open set in and consider a function f defined on Ω­ with values in S : where u = u ₀ and with We give differential operators as where and where Then the Coulomb operator (see [8] ) is Definition 2.1. Let Ω­ be an open set in . A function f ( z ) = f ₁ ( z ) + f ₂ ( z ) e ₂ is said to be an L ( R )- split hyperholomorphic function on Ω if the following two conditions are satisfied: In this paper, we consider a L-split hyperholomorphic function on Ω in . Let ξ ₀ = r cosh θ and ξ ₁ = r sinh θ with r ² = | zz *|. Then any z = ξ ₀ + Jξ ₁ can be expressed as z = r (cosh θ + J sinh θ ), where θ is the angle between the vector and the real axis. Theorem 3.1. Let ­Ω be a domain of holomorphy in . If u ( r , θ ) is a split quaternion function satisfying M ( D ) f = 0 on Ω­, then there exists a split hyper-conjugate quaternion function v ( r , θ ) satisfying M ( D ) f = 0 such that u ( r, θ ) + Jv ( r , θ ) is a split hyperholomorphic function on Ω. Proof. We put We operate the operator ∂ from the left-hand side of ϕ ( r , θ ) on Ω. Since and , we get ∂ ϕ ( r , θ ) is zero. Since Ω is a domain of holomorphy, the ∂-closed form ϕ ( r , θ ) is a ∂-exact form on Ω­. Hence, there exists a split hyper-conjugate quaternion function v ( r , θ ) satisfying M ( D ) f = 0 on Ω such that u ( r , θ ) + Jv ( r , θ ) is a split hyperholomorphic function on Ω. Example 3.2. If the split quaternion function in a domain of holomorphy is known, then a split hyper-conjugate quaternion function v ( r , θ ) of u ( r , θ ) on Ω can be found. That is, and f ( r , θ ) = u ( r , θ ) + Jv ( r , θ ) is a split hyperholomorphic function satisfying M ( D ) f = 0 on Ω. Theorem 3.3. Let Ω be an open set in and f be a split quaternion function satisfying M ( D ) f = 0 on Ω. T hen the multiplicative modulus of D f is Proof. For f = u + Jv and where Since and we have Theorem 3.4. Let f : → be a polar coordinates mapping defined by f(r, θ) = (r cosh θ, r sinh θ). Then the determinant of this mapping is where Proof. The chain rule gives Then Theorem 3.5. Let f : → be a polar coordinates mapping defined by f(r, θ) = (e^r cosh θ, e^r sinh θ). Then the determinant of this mapping is Proof. We can prove as above Theorem 3.4. Theorem 3.6. Let Ω be an open set in and f be a split hyperholomorphic function on Ω­. Then there exists a differentiable function φ on Ω such that the vector field Proof. We let any point on Ω. Consider where μ ( ξ ₁ ) is a split quaternion-valued function. By the fundamental theorem of calculus, we can find Since f is a split hyperholomorphic function on Ω and differentiating with respect to ξ ₁ , we obtain where and Putting and then we have