A contragenic function in a domain $\Omega\subseteq\mathbf{R}^3$ is a reduced-quaternion-valued (i.e. the last coordinate function is zero) harmonic function, which is orthogonal in $L^2(\Omega)$ to all monogenic functions and their conjugates. The notion of contragenicity depends on the domain and thus is not a local property, in contrast to harmonicity and monogenicity. For spheroidal domains of arbitrary eccentricity, we relate standard orthogonal bases of harmonic and contragenic functions for one domain to another via computational formulas. This permits us to show that there exist nontrivial contragenic functions common to the spheroids of all eccentricities.

中文翻译：

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球谐函数和球谐函数之间的关系

域 $\Omega\subseteq\mathbf{R}^3$ 中的逆向函数是一个约简四元数（即最后一个坐标函数为零）调和函数，在 $L^2(\Omega) 中正交$ 到所有单基因函数及其共轭。与和谐性和单基因性相反，矛盾性的概念取决于域，因此不是局部属性。对于任意偏心率的球体域，我们通过计算公式将一个域的谐波和逆向函数的标准正交基与另一个域相关联。这使我们能够证明所有离心率球体都存在共同的非平凡逆向函数。