Missundestanding of how complex numbers work not helping with fourier analysis

**emmv** · January 2nd 2010, 12:29 PM

I am studying electrical engineering at university level in the UK. As part of a course on Circuit and Signal processing I have tutorial sheets on Fourier Analysis.

I am fine with fourier analysis except when I start to look at exponential fourier series, I have the question, I even have the answers and how the tutor reached the answer (the workings). But I am not understanding how he is jumping from one line to another. I have attached the whole solution but the bit that I am misunderstanding is:

$ r_{m}=\frac{-V}{4\pi^2}\displaystyle\{\frac{-1}{jm}[2\pi]\displaystyle\}$

$ \\r_{m}=\frac{-jV}{4m\pi^2}\displaystyle\{[2\pi]\displaystyle\} $

where j = complex number $j^2=-1$ , how is j moving from the denominator to the numerator?

surely the next line should be

$ \\r_{m}=\frac{V}{j4m\pi^2}\displaystyle\{[2\pi]\displaystyle\} $

Any help would be greatly appreciated. Thanks Ed.

**CaptainBlack** · January 2nd 2010, 08:28 PM

Originally Posted by emmv

I am studying electrical engineering at university level in the UK. As part of a course on Circuit and Signal processing I have tutorial sheets on Fourier Analysis.

I am fine with fourier analysis except when I start to look at exponential fourier series, I have the question, I even have the answers and how the tutor reached the answer (the workings). But I am not understanding how he is jumping from one line to another. I have attached the whole solution but the bit that I am misunderstanding is:

$ r_{m}=\frac{-V}{4\pi^2}\displaystyle\{\frac{-1}{jm}[2\pi]\displaystyle\}$

$ \\r_{m}=\frac{-jV}{4m\pi^2}\{[2\pi]\} $

where j = complex number $j^2=-1$ , how is j moving from the denominator to the numerator?

surely the next line should be

$ \\r_{m}=\frac{V}{j4m\pi^2}\displaystyle\{[2\pi]\displaystyle\} $

Any help would be greatly appreciated. Thanks Ed.

Now multiply top and bottom by $j$ , this gives a $j$ on the top and on the bottom $j^2=-1$ , or just $-j$ on the top:

$ r_{m}=\frac{V}{j4m\pi^2}(2\pi)=\frac{Vj}{j^24m\pi^ 2}(2\pi)=\frac{-jV}{4m\pi^2}(2\pi) $

CB

**emmv** · January 3rd 2010, 03:29 AM

Thanks Captain Black, to clarify my understanding:

If I had $\frac{1}{5}$ and I multiplied top and bottom by 5 then the answer would be $\frac{5}{25}$ which is still $\frac{1}{5}$ the actual values haven't changed just the figures representing them. So is it the case that with $j$ that $j^2$ is $-1$ which effectively gets lost within the other numbers?

So actually the steps are as below, but then how are you moving the minus from the denominator to the numerator?

$ r_{m}=\frac{V}{j4m\pi^2}(2\pi)=\frac{Vj}{j^24m\pi^ 2}(2\pi)=\frac{jV}{(-1)4m\pi^2}(2\pi) $

Thanks, Ed.

**emmv** · January 3rd 2010, 04:07 AM

Sorry Captain Black, I didn't think before I asked my question. If the fraction was $\frac{1}{-5}$ this would in fact be $-\frac{1}{5}$ as the minus relates to the whole of the fraction. Thank you for helping me with this. Regards Ed.

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