bibkey: nist2026dlmfcomplexpowerhyperbolic authors: NIST Digital Library of Mathematical Functions year: 2026 title: Logarithms, Powers, and Hyperbolic Functions doi: null url: https://dlmf.nist.gov/4.2 claim: A selected logarithm defines a complex power; hyperbolic sine and cosine are exponential difference and sum. strata_touched: [] license: citation-only triage: anchor

Logarithms, powers, and hyperbolic functions

The primary definitions are DLMF §4.2(i), §4.2(iv), equation 4.2.28, and §4.28, equations 4.28.1–3.

For a selected logarithm value $L$ of a nonzero complex number $z$, the corresponding power is $e^{tL}$. DLMF’s principal power uses the principal logarithm and is analytic off the negative real cut; the two boundary values on that cut must be distinguished. In particular, for $\psi =-{\phi }^{-1}$ and $a=\log \phi$, the lower and upper values are $-a-i\pi$ and $-a+i\pi$. They give conjugate powers when $t$ is real, and equal powers when $t$ is an integer. These conclusions do not give pointwise conjugacy for nonreal $t$.

The hyperbolic definitions consumed by the theory volume are

\sinh z=\frac{e^z-e^{-z}}2,\qquad
\cosh z=\frac{e^z+e^{-z}}2,\qquad
\cosh z\pm\sinh z=e^{\pm z}.

The consumer is FIB hyperbolic geometry and phase boundary, §§一–三. The selected branch, matrix realization and five-mode source-law bridge are supplied by that consumer’s definitions and proofs; DLMF supplies the scalar definitions, not a native fractional tree operation or a physical interpretation.