Complex Fourier series step with exponential addition

[Jag1972]

Hello Folks,
I have a problem understanding a step of the complex Fourier series; it’s a step which involves simple addition and subtraction of exponentials (regrettably not simple for me).
I have attached a picture of the step I am having a problem with would really appreciate if someone could point me in the right direction.
Jag.

 

[uart]

Yes there is indeed an error in that text, and it's in the line immediately following the one you marked with an asterisk.

The line following the one you marked should read (for simplicity I'll just give the bracketed part of the expression) :

$$( \, \, 2\, \exp(j n \pi/2) - 2\, \exp(-j n \pi/2) - [ \exp(j n \pi) - \exp(-j n \pi) ] \, \, )$$

This is simply collecting like terms right. Note however that for integer "n" that exp(j n Pi) is equal to exp(-j n Pi) so the "[]" term on the end goes to zero leaving just :

$$(\, 2\,\exp(j n \pi/2) - 2\,\exp(-j n \pi/2) \, )$$

This is what should have been shown in the bracketed term of the line following the one you indicated. Note that the book has a typo where they used Pi instead of Pi/2, but the Pi/2 magically reappears correctly in the final answer that follows.

So essentially it's just a typo on the part of the book. Does that clear it up for you?

 

[Jag1972]

UART: Thank you very much I was pulling my hair out trying to work it out. I have attached my working out following your reply is this right (I have shown the trig identities). Thanks :)

 

[Jag1972]

UART: I have attached it as a JPEG as I think you may have to save the PDF to open.

 

[uart]

Almost right.

Where you wrote :

"where cos(q) = (exp(jq) + exp(-jq))/2"

it would have made a lot more sense if you had of written :

"where j sin(q) = (exp(jq) - exp(-jq))/2

 

[Jag1972]

UART: Thanks again I wrote the wrong identity down doh! Its got to be sin$$\theta$$

Thanks again UART :)