Efficiently Solving a System of Equations for Physics Problems

[Frank Einstein]

Homework Statement

Hi everybody, I am trying to find the solution to this system of equations for a bigger physics problem:

h+k+l=2n+1 with n=0,1,2,3...
e^{i(π/2)(h+k+l)}+e^{i(π/2)(3h+3k+l)}+e^{i(π/2)(3h+k+3l)}+e^{i(π/2)(h+3k+3l)}=0

Homework Equations

None

The Attempt at a Solution

Ihave tried to solve it by taking the common factor e^i(π/2) out and trying to solve
e^(h+k+l)+e^(3h+3k+l)+e^(3h+k+3l)+e^(h+3k+3l)=0 by using h+k+l=0; then, I arrive to 1+e^2k+2h+e^2h+2l+e^2k+2l=0.

From here I don't know how to keep going, so if anybody could point me what to do next it wolud be very helpfull.

Thanks for reading.

 

[haruspex]

Ihave tried to solve it by taking the common factor e^i(π/2) out and trying to solve

That is a factor of e^i(π/2)+x, not of e^i(π/2)x.

 

[RUber]

Homework Statement

Hi everybody, I am trying to find the solution to this system of equations for a bigger physics problem:

h+k+l=2n+1 with n=0,1,2,3...
e^{i(π/2)(h+k+l)}+e^{i(π/2)(3h+3k+l)}+e^{i(π/2)(3h+k+3l)}+e^{i(π/2)(h+3k+3l)}=0

To make these equal to zero, you need to enforce conditions on the periodic functions--both real and imaginary.
$\cos( \frac{ \pi(h+k+l)}{2} ) + \cos( \frac{ \pi(3h+3k+l)}{2} )+\cos( \frac{ \pi(3h+k+3l)}{2} )+\cos( \frac{ \pi(h+3k+3l)}{2} ) =0$
and
$\sin( \frac{ \pi(h+k+l)}{2} ) + \sin( \frac{ \pi(3h+3k+l)}{2} )+\sin( \frac{ \pi(3h+k+3l)}{2} )+\sin( \frac{ \pi(h+3k+3l)}{2} ) =0.$

From these, you should be able to say something about the cases where this must be true.

 

[RUber]

Think of these as :
$\cos( \frac{ \pi(h+k+l)}{2} ) + \cos( \frac{ \pi(h+k+l)}{2} +(h+k)\pi )+\cos( \frac{ \pi(h+k+l)}{2}+ (h+l)\pi )+\cos( \frac{ \pi(h+k+l)}{2}+(k+l)\pi ) =0$
and
$\sin( \frac{ \pi(h+k+l)}{2} ) + \sin( \frac{ \pi(h+k+l)}{2} +(h+k)\pi )+\sin( \frac{ \pi(h+k+l)}{2}+ (h+l)\pi )+\sin( \frac{ \pi(h+k+l)}{2}+(k+l)\pi ) =0.$

 

[Ray Vickson]

Homework Statement

Hi everybody, I am trying to find the solution to this system of equations for a bigger physics problem:

h+k+l=2n+1 with n=0,1,2,3...
e^{i(π/2)(h+k+l)}+e^{i(π/2)(3h+3k+l)}+e^{i(π/2)(3h+k+3l)}+e^{i(π/2)(h+3k+3l)}=0

Homework Equations

None

The Attempt at a Solution

Ihave tried to solve it by taking the common factor e^i(π/2) out and trying to solve
e^(h+k+l)+e^(3h+3k+l)+e^(3h+k+3l)+e^(h+3k+3l)=0 by using h+k+l=0; then, I arrive to 1+e^2k+2h+e^2h+2l+e^2k+2l=0.

From here I don't know how to keep going, so if anybody could point me what to do next it wolud be very helpfull.

Thanks for reading.

Please do not use all bold fonts for your solution; it looks like you are yelling at us!
Mod note: Removed excess bold font

Anyway, if you let $m = 2n + 1$, and if you note that $h+k+l = m$ implies
$$\frac{i \pi}{2} (3h+3k+l) = \frac{i \pi}{2} (3h + 3k + 3l - 2l) = \frac{i \pi}{2} 3m - i \pi l,$$
and so forth, you get a much simpler problem, especially if you multiply through by $e^{-3 m \pi i/2}$.

I would suggest you look at some of the initial cases $m = 1, 3, 5, \ldots$ separately, until you have gained sufficient insight into the nature of the solution for general odd $m$.

 

[Frank Einstein]

Thank you very much for your anwsers, guess I will be able to proper solve it.