To some extent you are right. What you think is fine for the functions having discrete Fourier spectrum. Then, indeed, everything is well described by two real series: one of multiple sinus functions and the other one consisting of multiple cosine functions, both with discrete (and the same) frequencies. Periodic functions have this property, imaginary unit i is not needed at all. But Fourier Transform is well defined for many other functions, including complex functions, using exactly one formula. Single formula is certainly easier to manage than two series, isn't it? In case of continuous spectrum it is much more convenient to work with complex amplitudes, which appear even after FT applied to nice real functions.

Marek Wojciech Gutowski "Single formula is certainly easier to manage than two series, isn't it?"

It is because, mathematics not developed in that way. In result, we have an imaginary number 'i', with this 'i' one can not give any meaningful-understanding to complex function or procedure or any complex outcome.

Respectful Deepak Bhalchandra Gode,

I'm looking at the example you study: (1+2i)/(2+3i). According to you, the result of this division, after somewhat complicated calculations is presented as:

0.615385 cos(A) + 0.076923 sin(A), where A = arc tan (0.615385/0.076923). Not bad approximation (but with unspecified accuracy), while 8/13 + i/13, also shown there, is an exact result, much easier to obtain by hand, without using trigonometric tables or so called scientific calculator.

You don't like imaginary unit, i. It makes complex numbers appearing always a sum of two parts. But how about two-dimensional vectors? Can you live with them?

You wrote: one can not give any meaningful-understanding to complex function. Really? Have you ever been talking with electrical engineer? He will explain to you what is amplitude, phase, phase shift and more.

Sir,

1) what I have shown in paper is just literal way, while considering complex number, radius R and angle thita is enough. Further simplification is possible.

Things must be shown correctly, the exact value of i is not defined but on the graph it is shown at length =exact one.

2) Interpretation of Factorial of real positive number ‘n’ and its gamma value is well known.

But what is meaning of (2+3i)!, its Gamma value is -0.082395273+0.091774287i also Gamma value of (5+3i) is 0.016041882-9.4332933i what it means actually?

3) as far as electricity (specifically AC) is concerned assumption of circle is more appropriate, but will write indetail on it, later.

To the question #2):

The domain of factorial is a set of natural numbers (plus zero), so the expression (2+3i)! simply doesn't make any sense. On the other hand, the Gamma function is defined on complex plane. One may think about Gamma as of continuous and differentiable extension of factorial, more or less, from natural numbers to complex domain. But did you notice that factorial is strictly increasing function while Gamma (on positive part of real axis) is not? Well, I don't see any obvious use of Gamma function with complex argument (contrary to factorial, very useful in statistics among other), but does it mean it is a nonsense? Mathematicians will certainly never agree with that.

"I don't see any obvious use of Gamma function with complex argument"

Correct, because one can not understand its functional relationship, also not explainable in real world.

Sir, what is your opinion or mathematicians opinion about complex number

1) it is a two dimensional number

2) it is a one and half dimensional number

3) its one dimensional number only

Complex numbers are just points on complex plane, exactly like better known points on any other Euclidean plane. What is then your answer about dimensionality in this second case?

To uniquely identify the point on ordinary plane we need its two coordinates. This is the same as to say that the plane is two-dimensional. The plane, not the points. Fractional dimensionality is only applicable to very special figures called fractals.

In real analysis the inegration of semicircle of radius r is equal to pi * r^2/2

Now I have same circle with specific position or location i.e. in complex number form

Integration of z over c, where c is a left side semicircle of a unit circle i.e. |z| =1 from z=-i to z=+i.

the solution of this integration is 2i

Now, what this answer '2i' means? How to interpret this answer?

Is complex number really a two dimensional?