The variation of induced EMF $\varepsilon$ with time $t$ in a coil if a short bar magnet is moved along its axis with a constant velocity The variation of induced EMF $\varepsilon$ 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."

 A: This is the waveform for a free falling, relative short length magnet through a long solenoid (Actual experiment):

The negative semi-period is larger in amplitude than the positive depicted above due to the g acceleration.
For an almost constant speed,  falling magnet the waveform looks like this bellow (Actual experiment):

Note: Due to constant speed the waveform is in this case symmetrical in amplitudes of positive and negative pulse.
In both cases the induced current is almost zeroed when total length of short magnet is inside the long solenoid. The flux change of the leading pole inside the solenoid is negated form the flux change by the trailing end pole of the magnet.
(see also this explanation https://physics.stackexchange.com/a/89296/183646).
In general for the pole of the magnet approaching one end of the solenoid at $vt$ distance from the solenoid opening (along the central axis of the long solenoid) and then leaving with its opposite pole the other end of the solenoid this formula holds:
$$I(t)=\frac{\mu_{0} q_{m}}{2 L}\left[1+\frac{v t}{\sqrt{(v t)^{2}+b^{2}}}\right]$$
with $b$ the inner radius of the solenoid, $t=0$ the time at which the dipole short magnet is totally inside the solenoid and the induced current is zeroed, $t<0$ the time in which the magnet is approaching the solenoid and $t>0$ the time in which it is leaving the other end of the solenoid. $L$ is the inductance of the solenoid, $q_{m}$ (in SI units of magnetic flux $Φ$, Wb) is the magnetic charge of the pole (using the magnetic pole model)  and $\mu_{0}$ the magnetic permeability of vacuum space. Use $q_{m}$ with an opposite sign for the magnet leaving the solenoid.
Then use,
$$\varepsilon=-L \frac{d I}{d t}$$
to find the self-induced emf voltage value.
