Is complex analysis more "real" than real analysis? In physics, in the past, complex numbers were used only to remember or simplify formulas and computations. But after the birth of quantum physics, they found that a thing as real as "matter" itself had to be described by complex wave functions and there's no way to describe it using only real numbers.
In mathematics, in real analysis, there's examples like the function $f(x)=\frac{1}{1+x^2}$, and why this function does not have the "smoothness" of the exponential function, polynomials, or the sine and cosine functions, why it has a radius of convergence equals 1 despite the fact that this function is infinitely differentiable. You can't see the reality of this function until you see it through the field of complex analysis, where you can observe that $f$ is not that smooth because it has 2 singularities in the complex plane.
I am just asking for examples like this such that when you see it in the narrow "window" of real analysis, you can't see the "reality" until you view it from the window of complex analysis. I am just starting to self learn complex analysis and I find it more natural than real analysis and it tells you the "truth" behind a lot of things.

One example is the two-dimensional electrostatics. The potential in the domain $D$ without electric charges satisfies the Laplace equation $\varphi_{xx}+\varphi_{yy}=0$. From the 'real' point of view, its solutions for


*

*two infinite uniformly charged planes

*two infinite coaxial uniformly charged cylinders
have nothing to do with each other. However, from the 'complex' point of view one is obtained from the other by the conformal transformation $w(z)=e^z$ which maps the strip $a<\Re z<b$ onto the annulus $ e^a<|z|<e^b$.
