We can always choose a small enough neighborhood
k' → 0
θ → π /2
so that a very large number of terms are needed in the forward formula, no matter how many iterations we apply. In this case, we integrate from the other end, selecting a change of variable which results again in a small parameter and angle.
Again, this is a placeholder. I will show how the formula is derived shortly.
Above, it is easy to see that by increasing m and pushing θ closer to the endpoint π /2, we can force the forward formula (here, carried out to 2 iterations) to require more and more terms. The conjugate arc, however, can be calculated in fewer and fewer terms as we approach the limit.
Being the thing to be done.