Log expansion for infinite solenoid

[Shinobii]

Hello, I found an approximation for this log function:

$$log \Bigg(\frac{\Lambda}{\rho} + \sqrt{1 + \frac{\Lambda^2}{\rho^2}} \Bigg),$$

where $\Lambda \rightarrow \infty$. The above is approximated to the following,

$$-log \bigg(\frac{\rho}{\rho_o} \bigg) + log \bigg(\frac{2 \Lambda}{\rho_o} \bigg).$$

How is this done? I tried expanding the $\sqrt{1 + x^2}$ 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 $\rho_o$ is, I assume some constant.

I found this approximation here:

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

 

[Shinobii]

Wow, never mind. Clearly I am being silly here, for $\Lambda \rightarrow \infty$.

$$log\bigg( \frac{\Lambda}{\rho} + \sqrt{1 + \frac{\Lambda^2}{\rho^2}} \bigg) \rightarrow log \bigg( \frac{ 2 \Lambda}{\rho} \bigg) \rightarrow log(2 \Lambda) - log(\rho).$$

As for the $\rho_o$ I have no idea why that enters the equation.