Tricks in Complex analysis, but not possible in Real analysis or Quaternion analysis

Question

1. What are some fundamental reasons that Complex analysis in $\mathbb{C}$ is more powerful, but it is not possible to generalize to the Real analysis in $\mathbb{R}$ or the Quaternion analysis in $\mathbb{H}$? What makes the $\mathbb{C}$ more special than the Real $\mathbb{R}$ or Quaternion $\mathbb{H}$?

2. What are some tricks that can be done in Complex analysis, but not possible in Real analysis or Quaternion analysis?

Here I provide additional details: for example in complex analysis, we have the Riemann zeta function. But there is no obvious counterpart in real analysis or in quaternion analysis.

Answer 1

One operation where complex analysis fits is tetration, which is defined for integer heights as $^2x=x^x$, $^3x=x^{x^x}$, etc. If we try to interpolate to define the operation for fractional "power-tower" heights, there are many ways to do it. But in complex analysis a unique interpolation with a single branch cut in the complex plane is defined:

It has now been proven[1] that there exists a unique function F which is a solution of the equation F(z + 1) = exp(F(z)) and satisfies the additional conditions that F(0) = 1 and F(z) approaches the fixed points of the logarithm (roughly 0.318 ± 1.337i) as z approaches ±i∞ and that F is holomorphic in the whole complex z-plane, except the part of the real axis at z ≤ −2. This proof confirms a previous conjecture.[2] The construction of such a function was originally demonstrated by Kneser in 1950.[3] The complex map of this function is shown in the figure at right. The proof also works for other bases besides e, as long as the base is bigger than ${\displaystyle e^{\frac {1}{e}}\approx 1.445}$. Subsequent work extended the construction to all complex bases. The complex double precision approximation of this function is available online.[4]

Cited references

1. Paulsen, W.; Cowgill, S. (March 2017). "Solving ${\displaystyle F(z+1)=b^{F(z)}}$ in the complex plane" (PDF). Advances in Computational Mathematics 43: 1–22. doi:10.1007/s10444-017-9524-1. S2CID 9402035.

2. Kouznetsov, D. (July 2009). "Solution of ${\displaystyle F(z+1)=\exp(F(z))}$ in complex $z$-plane" (PDF). Mathematics of Computation 78 (267): 1647–1670. doi:10.1090/S0025-5718-09-02188-7.

3. Kneser, H. (1950). "Reelle analytische Lösungen der Gleichung ${\displaystyle \varphi {\Big (}\varphi (x){\Big )}={\rm {e}}^{x}}$ und verwandter Funktionalgleichungen". Journal für die reine und angewandte Mathematik (in German). 187: 56–67.

4. Paulsen, W. (June 2018). "Tetration for complex bases". Advances in Computational Mathematics. 45: 243–267. doi:10.1007/s10444-018-9615-7. S2CID 67866004.