Complex-valued elementary-function reconstruction with EML-type operators

Determine whether a reconstruction of complex-to-complex elementary functions, even when restricted to the principal logarithm branch, is possible using an EML-type operator.

Background

The EML constructions described in the paper reconstruct real-valued elementary functions while permitting complex intermediate values. The authors note that extending this result to genuinely complex-valued input-output functions is a separate issue because the complex logarithm is multivalued and naturally defined on a Riemann surface.

The paper leaves unresolved even the principal-branch version of this complex reconstruction problem and suggests that a substantially different operator might be required.

References

Note that question whether similar reconstruction for $\mathbb{C} \to \mathbb{C}$ elementary functions, even reduced to principal branch, is possible, is separate and unanswered question.

— Diversity of EML-type operators  (2609.11210 - Odrzywołek, 10 Sep 2026) in Section 2.2, “Complex domain and branches”