A Robin boundary value problem for the Cauchy–Riemann operator in a ring domain

In this paper, we study a Robin condition for the inhomogeneous Cauchy–Riemann equation w z ¯ = f {w_{\bar{z}}=f} in a ring domain R, by reformulating it as a Dirichlet boundary condition.

The inhomogeneous Cauchy-Riemann equation for weighted smooth vector-valued functions on strips with holes

This paper is dedicated to the question of surjectivity of the Cauchy-Riemann operator $$\overline{\partial }$$ ∂ ¯ on spaces $${\mathcal {E}}{\mathcal {V}}(\varOmega ,E)$$ E V ( Ω , E ) of $${\mathcal {C}}^{\infty }$$ C ∞ -smooth vector-valued functions whose growth on strips along the real axis with holes K is induced by a family of continuous weights $${\mathcal {V}}$$ V . Vector-valued means that these functions have values in a locally convex Hausdorff space E over $${\mathbb {C}}$$ C . We derive a counterpart of the Grothendieck-Köthe-Silva duality $${\mathcal {O}}({\mathbb {C}}\setminus K)/{\mathcal {O}}({\mathbb {C}})\cong {\mathscr {A}}(K)$$ O ( C \ K ) / O ( C ) ≅ A ( K ) with non-empty compact $$K\subset {\mathbb {R}}$$ K ⊂ R for weighted holomorphic functions. We use this duality and splitting theory to prove the surjectivity of $$\overline{\partial }:{\mathcal {E}} {\mathcal {V}}(\varOmega ,E)\rightarrow {\mathcal {E}}{\mathcal {V}} (\varOmega ,E)$$ ∂ ¯ : E V ( Ω , E ) → E V ( Ω , E ) for certain E. This solves the smooth (holomorphic, distributional) parameter dependence problem for the Cauchy-Riemann operator on $${\mathcal {E}}{\mathcal {V}}(\varOmega ,{\mathbb {C}})$$ E V ( Ω , C ) .

PROPERTIES OF REGULAR FUNCTIONS OF A QUATERNION VARIABLE MODIFIED WITH TRI-COMPLEX QUATERNION

In a quaternion structure composed of four real dimensions, we derive a form wherein three complex numbers are combined. Thereafter, we examined whether this form includes the algebraic properties of complex numbers and whether transformations were necessary for its application to the system. In addition, we defined a regular function in quaternions, expressed as a combination of complex numbers. Furthermore, we derived the Cauchy-Riemann equation to investigate the properties of the regular function in the quaternions coupled with the complex number.