Log expansion for infinite solenoid

Shinobii

Hello, I found an approximation for this log function:

[tex] log \Bigg(\frac{\Lambda}{\rho} + \sqrt{1 + \frac{\Lambda^2}{\rho^2}} \Bigg), [/tex]

where [itex] \Lambda \rightarrow \infty [/itex]. The above is approximated to the following,

[tex] -log \bigg(\frac{\rho}{\rho_o} \bigg) + log \bigg(\frac{2 \Lambda}{\rho_o} \bigg). [/tex]

How is this done? I tried expanding the [itex] \sqrt{1 + x^2} [/itex] term, but I still don't get how they arrive to the above approximation.

Any help would be greatly appreciated!

Cheers!

I have no idea why this was sent to linear algebra section . . . And I do not know how to move it to classical physics. . .

jbunniii

What is ##\rho_0##? It appears in the second expression but not the first.

Shinobii

I do not know what [itex] \rho_o [/itex] is, I assume some constant.

I found this approximation here:

http://www.physicsforums.com/showthread.php?t=119419

Shinobii

Wow, never mind. Clearly I am being silly here, for [itex] \Lambda \rightarrow \infty [/itex].

[tex] log\bigg( \frac{\Lambda}{\rho} + \sqrt{1 + \frac{\Lambda^2}{\rho^2}} \bigg) \rightarrow log \bigg( \frac{ 2 \Lambda}{\rho} \bigg) \rightarrow log(2 \Lambda) - log(\rho). [/tex]

As for the [itex] \rho_o [/itex] I have no idea why that enters the equation.