Why Are w[n] = exp(j*pi*n) and w[n] = (−1)^n Equivalent?

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Discussion Overview

The discussion centers on the equivalence of two mathematical expressions: w[n] = exp(j*pi*n) and w[n] = (−1)^n. Participants explore the underlying reasons for this equivalence, focusing on the implications of integer values for n and the properties of exponential and trigonometric functions.

Discussion Character

  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant states that exp(i * pi * n) can be expressed using Euler's formula as cos(pi * n) + i*sin(pi * n).
  • Another participant assumes n takes on integer values and explains that for integer n, sin(pi*n) equals 0, leading to the conclusion that cos(pi*n) equals (-1)^n.
  • A different participant suggests that the equivalence can be shown by rewriting exp(j*pi*n) as (e^{j*pi})^n and noting that e^{j*pi} equals -1.

Areas of Agreement / Disagreement

Participants generally agree on the mathematical reasoning behind the equivalence, but the discussion does not resolve any deeper implications or alternative interpretations of the expressions.

Contextual Notes

The discussion assumes that n is an integer, which is crucial for the conclusions drawn about the sine and cosine functions. There is no exploration of cases where n might take on non-integer values.

Who May Find This Useful

This discussion may be useful for individuals interested in complex numbers, Euler's formula, or the properties of trigonometric functions in relation to exponential expressions.

perryben
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w[n] = exp(j*pi*n)

w[n] = (−1)^n

Hi, can anyone off hand verify why these two exressions are equal? Thanks
 
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exp(i * pi * n) = cos(pi * n) + isin(pi * n).

You figure out the rest.
 
I am assuming n takes on integer values. In which case and expansion of exp(i*pi*n)=cos(pi*n)+i*sin(pi*n). Now for n in Z, sin(pi*n)=0, since: sin(0)=sin(pi)=sin(2pi)=sin(3pi)=...=0. Similarily, cos(pi*n)=(-1)^n, since cos(0)=cos(2pi)=cos(4pi)=..cos(2kpi)=1, and cos(pi)=cos(3*pi)=...cos((2k+1)pi)= -1. Therefore exp(i*pi*n)=(-1)^n for any n in Z.
 
perryben said:
w[n] = exp(j*pi*n)
w[n] = (−1)^n
Hi, can anyone off hand verify why these two exressions are equal? Thanks

The easiest way is just to write [tex]e^{j \pi n} = (e^{j \pi})^n[/tex] and note that [tex]e^{j \pi} = -1[/tex]
 

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