\(e^{(-1+2j)t}\)+\(e^{(-1-2j)t}\)


my attempt:
\(e^{-t}e^{j2t}\) + \(e^{-t}e^{-j2t}\)

\(e^{-t}(e^{j2t}\) + \(e^{-j2t})\)

euler's identity says...

cos(2t)=\(e^{j2t}\)+\(e^{-j2t}\)/2

so...

\(e^{(-1+2j)t}\)+\(e^{(-1-2j)t}\) = \(e^{-t}(2cos(2t))\)

it seems right, but it's wrong! I checked the proof by substituting numbers into the original equation and the final equation and they evaluate to different numbers.

Any help would be nice.

 

jut,

Your solution is correct, but your substitution is wrong. I plotted both the given expression and your solution. They were identical.

Ratch

 

I plotted both the given expression and your solution. They were identical.

Thanks. OK the solution was correct. I must have goofed up the substitution last night.

Strangely enough, MATLAB simplified the expression to: 2*exp(-1)*cos(2*t)
which was very close.

 

jut,

Strangely enough, MATLAB simplified the expression to: 2*exp(-1)*cos(2*t)
which was very close.

I disagree with both MATLAB and your assessment. 2*exp(-1)*cos(2*t) = 2*0.37*cos(2t) is a constant amplitude sinusoidal wave. The correct solution is an exponentially damped sinusoidal wave.

Ratch