# I A question about the log of a rational function

1. Aug 7, 2016

### mmzaj

We have the rational function :
$$f(x)=\frac{(1+ix)^{n}-1}{(1-ix)^{n}-1}\left(\frac{1-ix}{1+ix}\right)^{n/2}\;\;\;,\;\;n\in \mathbb{Z}^{+}$$
It's not hard to prove that :
$$\frac{(1+ix)^{n}-1}{(1-ix)^{n}-1}=(-1)^{n}\prod_{k=1}^{n-1}\frac{x+i(\xi_{n}^{k}-1)}{x-i(\xi_{n}^{k}-1)}\;\;\;,\;\;\xi_{n}^{k}=e^{2\pi i k/n}$$
Now we want to compute $$\log f(x)$$ for x>0. The logarithm of the individual factors can be written as :

$$\log\left(\frac{x+i(\xi_{n}^{k}-1)}{x-i(\xi_{n}^{k}-1)}\right)=2i\tan^{-1}\left(\frac{x}{1-\xi_{n}^{k}}\right)+i\pi;\;\;\;\;x>0$$
So, one would expect:
$$\log f(x)=-in\tan^{-1}(x)-i\pi+2i\pi n+2i\sum_{k=1}^{n-1}\tan^{-1}\left(\frac{x}{1-\xi_{n}^{k}}\right)$$
But it looks nothing like what wolframalpha returns. What am i doing wrong here ?

Last edited: Aug 7, 2016
2. Aug 7, 2016

### micromass

Staff Emeritus
It might be helpful to say what wolfram alpha returns.

3. Aug 7, 2016

### mmzaj

graphically, there seems to be a difference between what i've calculated and the plot of $\log f(x)$ by multiples of $2\pi$ . but i am not able to locate the exact locations of the jumps.

4. Aug 7, 2016

### Staff: Mentor

You spent so much effort to type in post #1 and then you fail with some copies and pastes or links?
I hope micromass' crystal ball isn't out of order like mine currently is.

5. Aug 7, 2016

### mmzaj

have you ever used WF ? it doesn't return results for general n !! and posting one example won't be of help if it doesn't say where the jumps are ! thanks for the very helpful and constructive post anyways !!

6. Aug 7, 2016

### micromass

Staff Emeritus
If you refuse to give further information, then there is not much we can do. All I can say is that the complex logarithm is multivalued, so this can explain jumps of order $2\pi$.

7. Aug 7, 2016

### mmzaj

where should i expect the jumps to happen ? that's where i am stuck. and we can just forget about the graphical discrepancy and correct my analytic calculation.

8. Aug 7, 2016

### micromass

Staff Emeritus
Why don't you show us the graphics you got? I won't reply further to this thread if you don't show us what you did in wolframalpha.

9. Aug 7, 2016

### mmzaj

10. Aug 7, 2016

### mmzaj

it boils down to finding the discontinuities of $\log f(x)$