Complex number help!! (1 Viewer)

cchan334

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I don't know why but I keep doing this question but I keep getting the same wrong answer.
The question is:
Using Euler's formula, e^(ix) = cosx + isinx, show that if z=re^(ix), then:
1/z +1/(conjugate of z) = 2cosx/r^2

For some reason, I keep getting LHS = 2cosx/r instead of 2cosx/r^2
 

Trebla

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When you put over a common denominator use the fact that

 

cchan334

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When you put over a common denominator use the fact that

I tried that but I got the same answer? Maybe it's something wrong with my algebra?

I get LHS = 1/z + 1/(conjugate of z)
= z+ conjugate of z/ (mod of z squared)
= 2 Re(z) /r^2
= 2rcosx / r^2
= 2cosx/r ????
 

Trebla

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Ah right, didn’t work it fully through in my head. I think your answer is correct.
 

YonOra

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LaTeX, it’s embedded in this forum using [ tex ] and [ /tex ] wrapped around the code (without the spaces)
So once you click on latex and you have [ tex ] & [ /tex ] what do you do?
 

Trebla

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So once you click on latex and you have [ tex ] & [ /tex ] what do you do?
Put in the code for whatever you want to write up and post it. You can quote some of the posts above to see examples of what that code looks like.

There is also a guide in the stickied thread (located in the Maths forum) below (a bit outdated but most of it is still relevant)
 

YonOra

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Put in the code for whatever you want to write up and post it. You can quote some of the posts above to see examples of what that code looks like.

There is also a guide in the stickied thread (located in the Maths forum) below (a bit outdated but most of it is still relevant)
Cool, thank you
 

CM_Tutor

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So once you click on latex and you have [ tex ] & [ /tex ] what do you do?
So, for example, my post (excluding the tex codes) was:

\begin{align*} \frac{1}{z} + \frac{1}{\bar{z}} &= \frac{1}{z} \times \frac{\bar{z}}{\bar{z}} + \frac{1}{\bar{z}} \times \frac{z}{z} \\ &= \frac{\bar{z} + z}{z\bar{z}} \\ &= \frac{\bar{z} + z}{|z|^2} \text{ as } z\bar{z} = |z|^2 \\ &= \frac{2\Re{(z)}}{|z|^2} \text{ as } z + \bar{z} = 2 \times \Re{(z)} \\ &= \frac{2\Re{(z)}}{r^2} \text{ given } |z| = r > 0 \\ &= \frac{2 \times r\cos{x}}{r^2} \text{ given } z = r(\cos{x} + i\sin{x}) \\ &= \frac{2\cos{x}}{r} \end{align*}

Expanding this out...

\begin{align*}
\frac{1}{z} + \frac{1}{\bar{z}} &= \frac{1}{z} \times \frac{\bar{z}}{\bar{z}} + \frac{1}{\bar{z}} \times \frac{z}{z} \\
&= \frac{\bar{z} + z}{z\bar{z}} \\
&= \frac{\bar{z} + z}{|z|^2} \text{ as } z\bar{z} = |z|^2 \\
&= \frac{2\Re{(z)}}{|z|^2} \text{ as } z + \bar{z} = 2 \times \Re{(z)} \\
&= \frac{2\Re{(z)}}{r^2} \text{ given } |z| = r > 0 \\
&= \frac{2 \times r\cos{x}}{r^2} \text{ given } z = r(\cos{x} + i\sin{x}) \\
&= \frac{2\cos{x}}{r}
\end{align*}
  • the begin and end align* causes whatever character is preceded by an & to line up, putting the equal signs above each other
  • the \frac{x}{y} gives a fraction x over y
  • the function \bar{z} displays the conjugate of z
  • note that the braces { } are paired and so you must be careful to close each one you open
  • the \\ gives a new line (not needed in the line before the \end of align*
  • the functions like \Re{(z)} and \cos{x} treat Real part and cosine as functions and displays them
  • the \text{xxx yyy} inserts "xxx yyy" as text (including the space) instead of treating it as Maths and showing xxxyyy.
 

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