  We study the critical Ising model on the square lattice in bounded simply
connected domains with + and free boundary conditions. We relate the energy
density of the model to a fermionic observable and compute its scaling limit by
discrete complex analysis methods. As a consequence, we obtain a simple exact
formula for the scaling limit of the energy field one-point function in terms
of hyperbolic metric. This confirms the predictions originating in physics, but
also provides a higher precision.
