Do you know what is a complex exponential ???
This equation was developed by EULER . Usually referred to as the father of modern calculus he developed the first comprehensive approach towards complex numbers. He also derived the equation 
. The concept of the complex exponential has always been a confusing concept. We will try to give you a simplified approach. First we will try to understand the complex ‘i’ or ‘j’ used in the equation. As we all know that i= Ö-1 or i2=-1. This indicates a phase shift of 180 degree . So its understood that ‘i’ represents a phase shift of 90 degree. Any where in a equation it must be clear that i simply is a change in phase by 90 degree. For example if x=3+4i then it means that this is complex number with a magnitude of 3 on x axis and a magnitude of 4 on y axis since these axes are out of phase by 90 degree.
I=90 degree phase shift.
I square =180 degree phase shift.
I cube=270 degree phase shift.
I power 4=360 degree phase shift.
The rotation in anticlockwise direction is taken positive and clock wise is taken as negative by convention. So if we write 3-4i it simply means that this is a complex number with same magnitudes of x & y axis as the previous number used in the example expect that the rotation is now in the clockwise direction. Figure below explains this better.
Assuming now that the significance of the I or j is clear we can now proceed to understand the ‘w’ used in the complex exponential. W or omega is the angular frequency which comes into picture when we have rotational motion unlike linear motion where we use linear frequency denoted by ‘f’.
One thing has to be noticed that to represent a complex number we need a 2-D plane . To be precise any complex number is always a multidimensional number which need a minimum of 2 dimensions. All real numbers like natural numbers or rational numbers are all single dimensional. Notice that when we represent a complex number we have on the x & y axis which are just numbers I.e 3 & 4. But 
Is not as simple as any other complex number. The difference between a complex number like x+iy and Is the x & y axis. Is not a complex number but a complex function which has its x & y axis not numbers but functions I.e cosine & sine. Refer the figure below.
A COMPLEX NUMBER A COMPLEX FUNCTION
Keeping this in mind we can now say that the complex exponential is also a multidimensional function and we have to agree that a multidimensional number or a function is difficult to analyze and therefore its called complex. The complex exponential in 3 dimension looks the one shown below
Inspecting the complex exponential in 3 dimension we see that it’s a rotating helix. This helix moves to the right if its Ejwt & it moves to left if we have E-jwt. This function when seen in 2 dimension where we freeze the time axis looks like a rotating phasor with angular frequency of W. such a diagram in which we freeze the time axis and refer to only the phase changes is called the phasor diagram. From now we will consider ejwt as a phasor rotating in the anticlockwise direction & e-jwt as rotating phasor in the clockwise direction & it’s a practice to represent phasors with time axis suppressed. Using x-axis as coswt & y axis as sinwt the phasor diagram for Ejwt & E-jwt for different values of w is shown below. The black line represents Ejwt phasor which rotates in anticlockwise direction & red line is used for E-jwt which rotates in clockwise direction.
In unit 1 we have seen how fourier series helps us to find out the sines & cosines present in a signal. To compute the co-efficients we had to separately find out the sine co-efficient & then the cosine co-efficient & the DC component. The advantage of using a complex exponential is that it gives us the co-efficients of sines & cosines together as 
. If you have heard of negative frequency in signal analysis then you will get to know from where did it come to existence.
The very root of negative frequency concept is . Is a phasor rotating in anticlockwise direction with angular frequency w which is called the positive frequency. Similarly a phasor e^-jwt rotates with the same frequency w but in clockwise direction hence its called the negative frequency. The next unit will concentrate on the proof of Euler’s theorem i.e .
Points to remember
1. Any complex number is not one-dimensional and hence all complex
numbers are a minimum 2 dimensional.
2. A complex number has its x & y axis as number.
3. A complex functions has its x & y axis as functions.
4. The I or j simply represents a phase change of 90 degree
5. The complex exponential is a complex function with its x & y axis as functions I.e cosine & sine.
6. Ejwt is rotating phasor in anticlockwise direction & E-jwt is a rotaing phasor rotating in clockwise direction.
7. Ejwt in 3 dimensional view is a helix. This equation was developed by EULER . Usually referred to as the father of modern calculus he developed the first