The energy density in the planar Ising model

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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.