approximation of Ln within Tan

Sarah · **Posted:** Mon, 2 Mar 2009 10:21:22 UTC

I have a very complicated function where my variable x lies within a Ln that lies within a Tan.
I would very much like to have a function that could approximate it so I can estimate values of x.
I have no idea how to do it. does somebody have an idea?

aswoods · **Posted:** Mon, 2 Mar 2009 21:06:44 UTC

It really depends on the details of the function, and the range of values of x.

We could use

$\tan x = x + \frac13 x^3 + \frac{2}{15}x^5 + \frac{17}{315}x^7 + ...$

so (for example)

$\tan(\ln x) \approx (\ln x)+\frac13 (\ln x)^3 + \frac{2}{15}(\ln x)^5$

but this works only over the range 1/2 < x < 5/2

If the function is complex, you might be able to use

$\tan x = \frac{-i(e^{ix}-e^{-ix})}{e^{ix}+e^{-ix}}$

Are you looking for a polynomial?

Sarah · **Posted:** Thu, 5 Mar 2009 09:29:51 UTC

It's not that I especifically need a polynomial, I don't have anything

.
The hardest thing is indeed the Ln(x), I do not know how to work that there. I have only seen approximations for Ln(x+1) or very bad approximations for Ln(x)..

tashirosgt · **Posted:** Thu, 5 Mar 2009 16:52:37 UTC

The western novelist Louis L'Amour observed that beginning writers often waste pages and pages writing about their story instead of telling their story. Unless the game here to guess what function you are talking about, you could simply write it down. Also, in this age of computers, why do you need an approximation?

Sarah · **Posted:** Mon, 9 Mar 2009 11:44:13 UTC

Well, this is the reason....
$f(x)=1-\frac{y-z}{\sqrt{-(z-w)^2-4yx}}\mbox{Tan}\left[\frac{\sqrt{-\frac{1}{2}(z-w)^2-4yx}}{z+w}\mbox{Ln}\left(\frac{-yx}{z^2}\right)\right]=0$
I'm looking for the x...

tashirosgt · **Posted:** Tue, 10 Mar 2009 03:01:17 UTC

But it still isn't clear whether you need a numerical method (one like "bisection" that can be implemented as a computer program) to find x or whether you need symbolic expression for x.

Sarah · **Posted:** Wed, 18 Mar 2009 10:58:25 UTC

I would like a symbolic expression for x...

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approximation of Ln within Tan

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