interpretation :complex exponent ?

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HINDI

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Hi all


is there any interpretation for the complex exponent of e ?

what is that means?

salam
hindi
 

HINDI said:
Hi all


is there any interpretation for the complex exponent of e ?

what is that means?

salam
hindi
Click to expand...

\[e^z=cos(z)+jsin(z)\]
\[z=x+jy\]
 
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hi

i mean physical interpretation or any way but not a pure mathmatical interpretation .

see : Y^2 = Y*Y .

how can i explain e^j in the same way ?

salam
hindi
 

Have a look at Euler's Formula e, Pi, i, 1, 0 One zero. Complex numbers, Taylor series. Annual Conference, Mathematical Association - Antonio Gutierrez

Basically, \[e^{realnumber}\] means a regular exponent, increasing the distance of a point to the origin. e^0=1, e^1=e, ...
To double the distance of a point from the origin, we would multilply it with
\[e^{ln(2)}\]

\[e^{imaginarynumber}\] is a rotation with imaginarynumber radians.

\[e^{(a+bj)}\] is equal to \[e^a\dot{}e^{bj}\], meaning a scaling away from the origin by a factor \[e^a\] and a rotation over b radians.\]
 
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Hi all

the x & y axis are cos & sin while there is a 90 phase diffrence between them .

but :

why sin for im and cos for real not the opposite?

why always the e^j is related to rotation angle ?

why e when talking about rotation?

why j when talking about rotation?

all engineers study these formuls in thier unveristes , but they don't know the significant of these constants and the hidden mathematical meanings that lies there .




salam
hindi
 

See the power series expansion of \[e^{j\theta}, sin(\theta)\] and \[cos(\theta)\]. It will prove the Euler's formula.

why sin for im and cos for real not the opposite?
why always the e^j is related to rotation angle ?
why e when talking about rotation?
why j when talking about rotation?
Click to expand...
These are the results of the Euler's formula. First, you start from proving that Euler's formula is correct. Then, these are the results or interpretation from Euler's formula.
 
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Whenever I see e^jw, I visualize two sinewaves that are 90 degrees out of phase with each other (in quadrature), because that's how I implement them in analog or digital hardware. If you plot the two sinewave on an X-Y grid (some people call it I-Q or real-imaginary), you get a dot that revolves around the origin counter-clockwise (if positive frequency), or clockwise (if negative frequency). At any moment in time, the position of the dot tells you the signal's magnitude and phase, and those are extremely useful quantities when doing signal processing.

Personally I don't like e^jw notation, but it is very compact. Writing signal processing equations with sines and cosines quickly becomes messy and error-prone.
 

any complex number can be expressed in either of three forms:
1)cartesian: z=x+iy
2)polar: magnitud and angle.
3) exponential form which you are asking for.
the number preceding the "e" is the magnitude of the number, and the index is the phase angle of it.
 

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