The variation of induced emf (e) with time (t) in a coil if a short bar magnet is moved along its axis with a constant velocity

tack.imgur.com/hWpO5.png

can someone kindly tell the mathematical proof? Not the theoretical one stating Lenz law "As the magnet comes close, there will be emf induced, Later on, the magnet moves away, hence induced emf is present in opposite direction on compared to before."