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If $a_1=x$ (x is real) and $a_{n+1}=\Re(ln(a_n))$, where $\Re (z)$ is the real part of $z$, how does $a_n$ behave and what is $\displaystyle \lim_{n\to \infty} a_n$?

$a_n$ seems to jump around around in $(5,-5)$ for first 30 n values, does it ever stabilize - is there some bound on $a_n$ for large n?

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closed as not a real question by Ross Millikan, Akhil Mathew Apr 6 '11 at 4:44

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

    
re: $\sin y =y$ possible abstract duplicate math.stackexchange.com/questions/26750/… –  Please Delete Account Apr 6 '11 at 1:41
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It is quite unclear what is being asked here. Are $a_n$ supposed to be complex values? Presumably then, $\log(x)$ denotes the principal branch of the complex logarithm? What is $x$? Is it supposed to be a real number? Also, it would be nice if your title matched your question. –  JavaMan Apr 6 '11 at 2:18
    
The branch wouldn't matter if he swapped the order, as in $a_{n+1} = \Re(\ln (a_n))$. –  Carl Brannen Apr 6 '11 at 3:28
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Please don't change the question back and forth like this, as previously posted answers will look silly. Carl Brannen answered this version already. Voting to close, using "not a real question" despite that not being right. –  Ross Millikan Apr 6 '11 at 4:44
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I've closed this question. For reference, there was something of a rollback war, with the OP repeatedly changing their question (semantically). @solomoan: Feel free to open a new question, but please do not abuse edits in this manner. –  Akhil Mathew Apr 6 '11 at 4:47

2 Answers 2

up vote 6 down vote accepted

The other question, about the limit of $a_{n+1} = \ln(\Re(a_n))$, was deleted, but I'll solve it anyway.

It's a bit better to talk about this in terms of $b_n = \Re(a_n)$. Thus $b_n$ is real and $b_{n+1} = \Re(\ln(b_n)) = \ln |b_n|$. Let $f(z) = \ln |z|$. Now the only real fixed point, i.e. solution of $f(z) = z$, is approximately $-.5671432904$, so this (call it $z_0$) is the only possible limit of the sequence $b_n$. But $f'(z) = 1/z$, and since $|1/z_0| > 1$ that says the fixed point is unstable. That means we won't have a sequence $b_n \to z_0$ except in cases where some $b_n$ turns out to be exactly $z_0$.

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Now I see this question has migrated to another thread. Very annoying when questions won't stay still long enough to be answered. –  Robert Israel Apr 6 '11 at 4:37

Note that $\rm\ x\ =\ \log(\Re\ x)\ $ for ${\rm\ x\ =\ W(1)} + \pi\ i\ $ where $\rm\:W\:$ is the Lambert W function, which is worth being aware of for problems like this.

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in general should it be $\frac{W(-ln x)}{\ln x}$? it looks like the limit does not exist (over $\mathbb{R}$). Except perhaps in the case $log x =x$ .. (which does not have real solutions) –  Please Delete Account Apr 6 '11 at 3:16

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