Equation for underdamped harmonic motion

[pinkcashmere]

I found an explanation for the equation of under damped harmonic motion, x(t) = C cos(wt) + D sin(wt), but I was wondering if someone could further explain why:

- "However, if you assume the function x(t) is real, then they are related as A = B
- why is (A-B) is imaginary

 

[mfig]

If A and B are conjugates, A = (g,h) and B = (g,-h).

A + B = (g,h) + (g,-h) = (2g,0). This is a real number.
A - B = (g,h) - (g,-h) = (0,2h). This is a complex number.

 

[pinkcashmere]

If A and B are conjugates, A = (g,h) and B = (g,-h).

A + B = (g,h) + (g,-h) = (2g,0). This is a real number.
A - B = (g,h) - (g,-h) = (0,2h). This is a complex number.

thanks,
can you also explain the "However, if you assume the function x(t) is real, then they are related as A = B" bit