Simple complex power: why is e^( i (2*Pi*n*t)/T ) not 1?

In summary, the complex form of the Fourier series is a mathematical representation of a periodic function that can be expressed as a sum of complex exponential functions. However, when looking at one of the terms, e^[iωnt], it can be simplified to just c, since e^[iωnt] = 1 for all integer values of n. This simplification may seem odd, but it is due to the nature of complex numbers and their properties, such as non-uniqueness in taking fractional powers. This can lead to unexpected results, such as 0 for all values of t in this case.
  • #1
Aziza
190
1
the complex form of Fourier series is:

f(t) = Ʃ c*e^[iωnt]
where c are the coefficients, the sum is from n= -inf to +inf; ω= 2*pi/T, where T is period...

but if you just look at e^[iωnt] = e^[ i (2*pi*n*t)/T] = {e^[ i (2*pi*n)] }^(t/T)

where I just took out the t/T...
well, e^[ i (2*pi*n)] = 1, since n is integer...and (1)^(t/T) is still equal to 1...so shouldn't the complex Fourier form just reduce to f(t) = Ʃ c ?

I feel i must be doing something stupid, if someone could just please point out what exactly...
 
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  • #2
Every complex number, except 0, but including 1, has n distinct nth roots.
When dealing with complex numbers, 1 to a fractional power is not just 1.
 
  • #3
In general, e^(a*b) != (e^a)^b with complex numbers a,b - unless you care about the phase of the expression in some other way.
 
  • #4
HallsofIvy said:
When dealing with complex numbers, 1 to a fractional power is not just 1.

Nice one!
I'm just realizing that ##1^\pi## is the complex unit circle! :)
 
  • #5
Hello Aziza,
In case you're still skeptical, here's a couple of examples. If you had something like:

eiπ/3 = (e)1/3 = (ei/3) = 1/2+sqrt(3)/2.

Then the identity applies, but take a look here:

(e2πi)i = 1i =/= e-2π = e2∏ii

The identity does not hold, and you can't really guess when and where it does, or doesn't.
In your case, you know it doesn't work because you get such an odd result, 0 for all t,and 00 for t=0, when we know for a fact that eiωnt are n rotating vectors in the complex plain!
 
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1. Why is e^( i (2*Pi*n*t)/T ) not 1?

The reason why e^( i (2*Pi*n*t)/T ) is not 1 is because it is a complex number and follows the rules of complex arithmetic. In this case, the exponent contains the imaginary unit i, which when raised to a power greater than 1, results in a complex number. This complex number has a magnitude of 1, but its angle or phase is dependent on the values of n, t, and T. Therefore, it cannot be simplified to 1.

2. What is the significance of e in e^( i (2*Pi*n*t)/T )?

The constant e is a mathematical constant that is approximately equal to 2.71828 and is the base of the natural logarithm. In this context, e is being raised to a complex power, which results in a complex number with a magnitude of 1. This complex number is used in many mathematical and scientific applications, including signal processing and quantum mechanics.

3. How does the value of n affect e^( i (2*Pi*n*t)/T )?

The value of n in e^( i (2*Pi*n*t)/T ) determines the angle or phase of the complex number. As the value of n increases, the angle of the complex number also increases. This means that the complex number will have a different phase or position on the complex plane, but its magnitude will still be 1.

4. What is the role of t and T in e^( i (2*Pi*n*t)/T )?

The variables t and T represent time in the equation e^( i (2*Pi*n*t)/T ). T is the period or length of one cycle, while t is the specific time or position within that cycle. The values of t and T determine the frequency of the complex number and can also affect its phase.

5. How is e^( i (2*Pi*n*t)/T ) used in science?

e^( i (2*Pi*n*t)/T ) is used in various scientific fields, including physics, engineering, and mathematics. In physics, it is used to represent the periodic behavior of physical phenomena, such as electromagnetic waves. In engineering, it is used in signal processing and control systems. In mathematics, it has applications in complex analysis and Fourier analysis. Overall, e^( i (2*Pi*n*t)/T ) is a powerful tool in understanding and analyzing complex systems and phenomena.

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