Hello i'm new over here Hope i'm writing where it should be

[abel_ghita]

hello! I'm new over here.. Hope I'm writing where it should be written.
What is the solution of
x=tangent(x)
?
Well.. in fact I'm looking for the point where the graphs of function tangent and its inverse, arctangent, intersect each other (if we plot them on the same coordinated frame)... These three points of intersection must be on the first bisector, as we know that these two functions are symmetric with respect to the first bisector.
Thanks!

 

[micromass]

Well, 0 is an obvious answer. But I don't think you'll find the other two points explicitely. This is not unusual, the equation x=cos(x) also has no known explicit solution.

However, you can very easily approximate the answer by Newton-Rhapson or similar algorithms...

 

[abel_ghita]

Thanks!
Yes..those two other solutions were my problem...Newton-Rhapson?? i didn't even hear about this so far.. but i'll search for it... In the mean time.. could you find this approximation for me?
Thanks!

 

[abel_ghita]

Thanks!
Yes..those two other solutions were my problem...Newton-Rhapson?? i didn't even hear about this so far.. but i'll search for it... In the mean time.. could you find this approximation for me?
Thanks!

http://www.wolframalpha.com/input/?i=x=arctan(x)&lk=4&num=2

so I guess +-1.5708

 

[Bacle2]

Are you sure? Pi/2 and Tangent(Pi/2) are not close... (and notice what happens

at Pi/2 )

 

[Mute]

Funny that this was revived by the OP after more than a year since the last reply.

At any rate, $x = \arctan(x)$ only has one real solution, x = 0. You can prove this by looking at the derivatives: the derivative of x is of course 1, and you can show that away from x = 0 the derivative of arctan(x) is always less than one, so there will not be any intersections between the two curves except at x = 0.

However, an equation like $x = \arctan(2x)$ does have three solutions.

A similar equation, $m = \tanh(2z\beta J m)$, arises when solving the mean field Ising model for the magnetization m, demonstrating that there exists a phase transition from a paramagnet, corresponding to m = 0, to a ferromagnet, $m \neq 0$.