In complex analysis, every harmonic function admits a decomposition as a sum of a holomorphic and an anti-holomorphic function. However, this fact does not hold for paravector-valued harmonic functions, or so-called 𝒜-valued harmonic functions, in quaternion function theory. In previous articles, the authors proved that by taking into account ψ-hyperholomorphic functions, where ψ is a structural set different from the standard one and its conjugation, every 𝒜-valued harmonic function can be written as a sum of a monogenic, an anti-monogenic and a ψ-hyperholomorphic function. In this paper, we revisit such a decomposition and apply it to study problems in elasticity.
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