In order to examine the differentiability of a real function at a point x0, one has only two basic directions to move; in particular, to the left and to the right of x0. This is expressed in the property that if the two one-sided limits of a real function f at x0 exist and are equal, then the limit of f at x0 exists and is equal to either of the two one-sided limits.

On the other hand, in order to examine the differentiability, and by extension the analyticity, of a complex function at a point z0 of the complex plane, one has infinitely many directions to move; in particular, there exist the two previous directions; i.e. to the left and to the right of z0, plus rotations of the given point z0 around another point of the complex plane.

It may be the case that the value of the function changes after a 360-degree rotation of z0 around another point. For instance, considering the square root of a non-zero point z0 on the complex plane, a 360-degree rotation of z0 around the origin leaves z0 unchanged; i.e. z0->z0, but its square root changes sign, since, in polar form, z0=abs(z0)exp(iφ), φ being the argument of z0; hence sqrt(z0)=sqrtabs(z0)exp(iφ/2) and then when φ->φ+2π, we have that exp(iφ/2)-> exp(iφ/2) exp(iπ)=- exp(iφ/2). This property that says that the square root function has a branch point at 0 is “invisible” on the real line.

For more on this, you may find the following link useful:

Presentation A Physicist’s Look at the Multi-Valued Complex Functions: Th...

Another point is that, by definition, a complex function is analytic at a point z0 if it is differentiable not only at z0, but also in a whole neighborhood of z0. A consequence of this is that a one-time differentiable complex function is infinitely many times differentiable, which does not hold for real functions.

For a simple explanation of this, you may find the following link useful:

https://math.stackexchange.com/questions/640/why-are-differentiable-complex-functions-infinitely-differentiable

Complex Variables... the subtitle is "Practical Applications" You can read a good portion of the book by clicking on "Look Inside". All of the software for the examples is always free at the link in the Forward. I have used complex variables to solve several practical problems, including a massive environmental cleanup. https://www.amazon.com/dp/B07MW6782P

Let's take a concrete example to clarify the idea : Suppose that as an engineer you intend construct a fan for a computer to cool it down. Then, you better know how the heat spread.

Therefore, you consider the map u: R^2--->R for which for each (x,y) in R^2, u(x,y) denote the temperature at the point of coordonnate (x,y) on the computer (it is more accurate to say the variation of temperature by the way).

Because of some physical principles, u must satisfy the heat equation. It is therefore known that u must be analytic.

When you then consider the harmonic conjugate of u, you have a harmonic map f : R^2--->R^2. Who have thus two main way to view it : either you study it just as map of the real vector space R^2, or you can see it as a complex variables map.

It turns out that the second view point has a lot of interest because of all the miraculous theorems which appears in this world. Therefore, you can get from it a lot of qualitative applications that just approximations cannot give you : the power series expansion alone can lead to a very slow convergence which is not necessarely pratical.

An of course, with the qualitative properties one can understand much more than by just plotting a function. A good example of this is the mean value theorem which will basically tell us that the solution of some physical problem must behave such that the value at each point take care of what happen around this point.

There are three importent and interest problem in Complex Analythis:

1. Four-dimensial graphs of complex function.

2 Visualisation of all complex roots of polynom on the complex plane and migration of theese ropts with varying of right side

For example, my articles "Visualization of graphs of complex functions and

exact geometric method of finding complex roots of

a polynomial",

3. Application of complex numbers in economic, for example,

"Calculation of Cash-Flow Based on the Complex Interest Rate"

To begin the argument, the Euler's formula.

To see first apparitions of complex numbers, history of cubic equations shows Rafael Bombelli solving the "sophisticated" cases using the method of del Ferro/Tartaglia, when a temporary couple of imaginary numbers and it seemed that the formula/method was not applicable.

So, initially, complex numbers were useful to have working algorithms.

Then they obtained the geometric representation of the complex plain.

Then new formulas were extended using complex numbers by (Euler and many others).

Whole classes of functions were described in a more complete way in C than in R: polynomials and analytic functions, holomorphic functions, meromorphic functions, series of functions (and , by extension, of numbers: extensions of summability and so on).

In Systems Engineering, linear systems can be modeled by transfer functions, that are complex functions of the aforementioned functions. See Bode plot, too.

There are many other arguments, but others can provide.

I would prefer to conclude going back to arithmetic: Riemann Zeta Function, extended to C, is about some important properties of the distribution of prime numbers.

Applications of the prime numbers? To begin: cryptography, information security.

Thank you all