Let \(\mathbb{D}\) be the two-dimensional real algebra generated by 1 and by a hyperbolic unit k such that \(k^{2} = 1\). This algebra is often referred to as the algebra of hyperbolic numbers. A function \(f : \mathbb{D} \rightarrow \mathbb{D}\) is called \(\mathbb{D}\)-holomorphic in a domain \(\Omega \subset \mathbb{D}\) if it admits derivative in the sense that \({\rm lim}_{h\rightarrow{0}}\frac{f({\mathfrak{z}_{0}+h)} -f{(\mathfrak{z}_{0})}} {h}\) exists for every point \(\mathfrak{z}_0\) in \(\Omega\), and when h is only allowed to be an invertible hyperbolic number. In this paper we prove that \(\mathbb{D}\)-holomorphic functions satisfy an unexpected limited version of the identity theorem. We will offer two distinct proofs that shed some light on the geometry of \(\mathbb{D}\). Since hyperbolic numbers are naturally embedded in the four-dimensional algebra of bicomplex numbers, we use our approach to state and prove an identity theorem for the bicomplex case as well.

中文翻译：

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双复函数和双曲变量的函数的几何和恒等式定理

令\（\ mathbb {D} \）是1和双曲单元k生成的二维实数，使得\（k ^ {2} = 1 \）。该代数通常称为双曲数的代数。函数\（F：\ mathbb {d} \ RIGHTARROW \ mathbb {d} \）被称为\（\ mathbb {d} \）域中-holomorphic \（\欧米茄\子集\ mathbb {d} \）如果它从\（{\ rm lim} _ {h \ rightarrow {0}} \ frac {f（{\ mathfrak {z} _ {0} + h）} -f {（\ mathfrak {z } _ {0}）}} {H} \）存在的每个点\（\ mathfrak {Z} _0 \）在\（\欧米茄\） ，和当ħ仅允许为可逆双曲数。在本文中，我们证明\（\ mathbb {D} \）-全纯函数满足恒等式的一个出乎意料的有限形式。我们将提供两个不同的证明，以证明\（\ mathbb {D} \）的几何形状。由于双曲数自然地嵌入在双复数的四维代数中，因此我们使用我们的方法进行陈述并证明双复数情形的恒等式定理。