How to Simplify This Equation with x=z+1?

[ComFlu945]

I'm trying to follow some notes in class.

so the original equation is:
u= [ (x^2 – 1)^1/2 + ln (x + (x^2 – 1)^1/2 )]

and we let x=z+1.

Then by simplifying and approximating, u=z*2*2^.5 + higher order terms

from my notes: (x^2 – 1)^1/2 = (2z)^.5*(1+z/2)^.5
ln(1+e) approximately = e

 

[CompuChip]

from my notes: (x^2 – 1)^1/2 = (2z)^.5*(1+z/2)^.5
ln(1+e) approximately = e

I presume those are the two lines that have you confused?

The first one is most easily seen by noting that
x² - 1 = (x + 1)(x - 1)
which, if z = x - 1, is z(z + 2). Then the algebra is straightforward (take out a factor of 2 and split the root of product into product of roots).


For the second one, you can make a Taylor series expansion around x = 0 of
ln(1 + x) = ln(1) + 1/(1 + 0) x + O(x²)
and see that to first order, you get x.