What is the significance of -Mu in equation equality and boundedness?

[ssky]

http://dc191.4shared.com/img/6pyFHiMb/s7/0.8584304740152441/706869958.jpg

I need more explanation for the first equation in the picture above:

Why we said that each side in the equation is equal (-Mu) ?

 

[tiny-tim]

hi ssky!

(have a mu: µ )

the LHS is the same for all x, and the RHS is the same for all t,

and since they're equal, they must be the same for all x and t

 

[ssky]

:shy:

I am sorry,
can you explain more.
and why we took the negative value?

 

[tiny-tim]

the equation has to be true for all values of x and t

the LHS is a function of t, but for a fixed value of t it is the same for all values of x …

if you were to plot it on a 3D graph, with horizontal x and t axes, it would be a hillside on which all the contour lines (lines of level height) were parallel to the t axis…

but the RHS would be a hillside on which all the contour lines (lines of level height) were parallel to the x axis …

but the LHS and the RHS have to be the same hillside …

the only way that can happen is if the hillside is completely flat! ​

and why we took the negative value?

without seeing the previous page (which i don't want to ), I've no idea

 

[ssky]

Actually,

i didn't understand



can you give me a clear example about this problem?


​

 

[Office_Shredder]

The equation looks like this:

f(t)=g(x)

The left side is a function of t only, and the right side a function of x only. Now, pick t=0.

f(0)=g(x)

This is true for ANY value of x, so it must be that g(x) is a constant function. Different values of x don't change g(x), because you always get f(0).

Similarly, if we set x=0 we get
f(t)=g(0)

This is true for every value of t, so f(t) is a constant function as well

 

[HallsofIvy]

:shy:

I am sorry,
can you explain more.
and why we took the negative value?

It says in what you posted "where the negative value was forced to warrent the boundedness of the function \Gamma(t) as t\to \infty".

I suspect that if you set it equal to "a" where a could be any constant, you would eventually get a function involving e^{at} which will go to infinity if a is positive. Writing a= -\mu^2 where \mu can be any real number forces a to be negative so that e^{at} does NOT go to infinity and is bounded.