Without prior knowledge on the basics of complex numbers, this problem is impossible to solve. For example, it must be known that ~i = \sqrt{-1}~ and that the rectangular coordinate form of a complex number is ~z = a + bi~.

For the first subtask, since ~x \le 1~, we can utilize a simple if statement. If ~x = 0~, the solution is ~1 + 0i~. If ~x = 1~, the solution is simply ~0 + 1i~.

For the second subtask, some quick Googling will give the solutions to when ~x = 2~ and when ~x = 3~. When ~x = 2~, the solution is ~e^{-\pi/2} \approx 0.20788 + 0i~. When ~x = 3~, the solution is ~e^{\pi i e^{-\pi/2}/2} \approx 0.94716 + 0.32076i~. Alternatively, one can solve for when ~x = 2~ and ~x = 3~ manually using Euler's formula.

For the third subtask, one can utilize a for loop that iterates from ~1~ to ~x~. Start with ~2~ floating-point variables, ~a_0~ set to ~1~, and ~b_0~ set to ~0~. At each iteration of the for loop, it can be found that:

$$\displaystyle \begin{align*} a_i &= e^{-\pi b_{i-1}/2} \cos \frac{\pi a_{i-1}}{2} \\ b_i &= e^{-\pi b_{i-1}/2} \sin \frac{\pi a_{i-1}}{2} \end{align*}$$

Alternatively, one can utilize a complex numbers library in their preferred programming language.

For the last subtask, one can figure out the solution converges to approximately ~0.4383 + 0.3606i~, which means that after a certain ~x~, the solution becomes consistent up to ~4~ decimal places. This means that we can hardcode a value instead of looping up to ~x~. This certain ~x~ and the proof is left as an exercise for the reader.

Time Complexity: ~\mathcal O(x)~