Geometric Series

[tshort]

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Hi again, quick question regarding (a+bi)/(c+di)

What is the justification for dropping the imaginary part? i.e. Only using (ac+bd)/(c^2+d^2) and dropping the i[(ac+bd)/(c^2+d^2)] component

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Original equation:

2 * Sum[ (1/3)^t cos(wt) ]

You then perform a manipulation trick adding in (1/3)^t sin(wt) so it is summable by geometric series (2 * Sum[ ((1/3) e^iw)^t]). This equation is a new equation different from the original one. You have added an imaginary part to the equation so you may sum the series. You then drop the imaginary part after you sum the series and are left with the equal real parts.

On the other hand, you may use the identity:

cos(z) = (e^(iz)+e^(-iz)) / 2

Then your series becomes:

Sum[(1/3*e^(iw))^t] + Sum[(1/3*e^(-iw))^t]

Which sums to:

-(e^iw)/((e^iw)-3) + 1/(3e^(iw)-1)

Algebraic manipulations will then yield your desired equation (using (e^iw = cos w + i sin w):

1-3Cos(w)/(3Cos(w)-5)

In this second method you don't have to worry about adding in an imaginary part and later dropping it.