Derivation of output voltage of a Colpitts Oscillator
Toshiba's paper doesn't need any random poetry add-on, so forget the idea to mix it randomly with an obscure op-amp catch from the web sea.

Toshiba's paper gives a clear formula for the loop gain v2/v1. I would test it with simulation to be absolutely sure. Or recalculate it. Recalculation is = solve v2 with usual complex phasors and node voltage equations. The current generator outputs current Gm * v1 to passive circuit rd, C1, C3, L2. The circuit needs no equation solving, it's a 2 stage voltage divider. See NOTE2.

Gain v2/v1 is the formula of the output voltage v2 divided by v1 i.e. factor v1 simply removed.

The loop gain should be real exact +1 for pure sinusoidal oscillation. In practice it must be a real number > +1 to get the oscillation started. The amplitude will gradually grow until the signal gets distorted. 

To get the oscillation frequency you must solve from the equation of v2/v1 the frequency which makes v2/v1 real. It's elementary complex number algebra (see NOTE1) if the capacitors and inductors are so large that the frequency is so low that the phase lag caused by the slowness of the fet is neglible. Otherwise also Gm should be taken as complex.

NOTE1: The imaginary part of the denominator should be zero, The gain of the fet should be high enough to compensate the attenuation caused by Rd and the real part of the denominator.

NOTE2: Current source -Gm * V1 feeds an impedance, say Za, which has 3 parts in parallel. The parts of Za are 1)Rd, 2)C1 and 3)L2 &amp; C3 in series. The voltage over Za = -Gm * V1 * Za.

The final feedback to the fet i.e. V2 = the output from the voltage divider made of L2 and C3. The input to that voltage divider is the voltage over Za.