By default I think quad calls qagse which does support singularities, you can also specify a series of domains to avoid them. With the right set of options you can get to most of the quadpack routines.
You need to include the cross terms in the outer product. The density matrix for your pure state ##|\psi\rangle## is ##\rho= |\psi\rangle\langle\psi|## which includes terms like ##|00\rangle\langle 01|##
For part (ii) of task 1 you want to write ## y = av_1 + bv_2 + cv_3 ## and find ##a##, ##b## and ##c##. What happens if you e.g. take the inner product of this equation with ##v_1##?
Not quite. The function g(t) is even but you are integrating g(t)e^{j\omega t} which isn't even, so you need to do the other half of the integral, or notice that the negative time part is the complex conjugate of the positive time part.
The Fourier transform of a Lorentzian isn't a Lorentzian (its a decaying oscillation)
The Fourier transform of a Gaussian is a Gaussian, which is I guess what you mean?
Do you know any theorems about the Fourier transform of a derivative to help answer your other question?
If you do something like
fullscreen = get(0,'ScreenSize');
set(0, 'DefaultFigurePosition', [0 -50 fullscreen(3) fullscreen(4)])
before plotting anything then it will set the default size for new figures to be maximised.
Hope that helps :-)
Not quite, this is why the constant of integration is important. The first integration gives
\frac{df}{dx}=\frac{x^4}{4}+5x+c_1
and the second integration gives
f(x)=\frac{x^5}{20}+\frac{5x^2}{2}+c_1x+c_2
Which is why we need one constant of integration for each integral