The Analytical Solutions of Dyonic Black Holes in Einstein-Euler-Heisenberg Theory

Recently, we constructed analytical purely electric and purely magnetic black-hole solutions in Einstein–Euler–Heisenberg theory, while the corresponding dyonic solutions were obtained numerically \cite{Luo:2026srx}. Motivated by the recent analytical construction of a dyonic black hole at the special coupling locus $b=a\mathrm{/}2$ \cite{Ahmed:2026ufj}, we revisit the general dyonic sector. We show that exact analytical dyonic black-hole solutions can be constructed for nonlinear couplings satisfying $2b>a$, without imposing the restriction $b=a\mathrm{/}2$. The mass function is expressed in closed form in terms of the Lauricella hypergeometric function. In particular, our construction yields an exact dyonic solution for the Euler–Heisenberg coupling $b=7a\mathrm{/}4$. The solution of Ref.~\cite{Ahmed:2026ufj} is recovered in the limiting case $2b\to a$.