# Transcendental equation problem

1. Aug 13, 2015

### dji3214

i have a transcendental equation and i have not a mathematique superieur formation ( i'm an hydraulic engeneer) and i want to resolve it but i can't so if you can help me with it !!

the equation is : 2*x*n*ctg(2x)= x2 - n2 or (same equation) : (n*ctg(x)-x)*(n*tg(x)+x) =0

n= constante ( i have this n.... in example n = 0.5 )
xp = are the roots of equation above

2. Aug 13, 2015

### Staff: Mentor

Welcome to the PF.

Did you mean to put this in the schoolwork forums? If it's for your work, it could be posted in the general Math forums...

3. Aug 13, 2015

### SteamKing

Staff Emeritus
Such equations are normally solved by iteration, also known as trial and error.

4. Aug 13, 2015

### Ray Vickson

First: the $\cot(2x)$ function "blows up" (goes to $\pm \infty$) at (infinitely many) finite values of $x$, so that will give trouble in any numerical solution scheme. It is better to eliminate that problem by re-writing the equation as
$$2 n x \cos(2x) = (x^2 - n^2) \sin(2x)$$
Now a plot of both sides will reveal the approximate positions of roots, and these can be refined by various methods to get accurate values. There are infinitely many positive and negative roots.

5. Aug 13, 2015

### dji3214

DEAR ,, RAY thnx for helping me,, if i understand u this infitely positive and negative roots are for this n in our example 0.5 .... AND if i tell you that xp the roots are depending to this index p like in series Σ ...because in my formula they wrote in it Xp like this ; X is h ( hn = are the roots of equation above) ,, the index p is n and n (n=0.5) in the equation is mew !!

the GENERAL EQUATION is ;

where in the left side constant i have it

the right side this serie and THIS INDEX n batherd me a lot and complicate my equation because i dont know in wich number of n i stop my serie and also stop the root of the transcendental equation

Last edited: Aug 13, 2015
6. Aug 13, 2015

### Ray Vickson

I cannot understand anything you have written.

7. Aug 14, 2015

### dji3214

lol ,, ok ,, my general formula is : my problem is hn roots of transcendental equation

in PHI n i have hn ( hn is roots of the first transcendental equation that i wrote in first time )

this is PHIn

look at hn ,, now hn are roots of the transcendental equation that u transform to 2nxcos(2x)=(x2−n2)sin(2x) i would tell you that the greek lettre eta ( ) is n in first post thats all because i change variables lettre

Last edited: Aug 14, 2015
8. Aug 14, 2015

### SteamKing

Staff Emeritus
η is called "eta"

μ is called "mu", or "mew", as you prefer.

9. Aug 14, 2015

### dji3214

lol see i forget very think lol