Green's Functions

The Green's functions are given by the Fourier transform of the corresponding lineshapes. For a Lorentzian lineshape, the Green's function is

\[ G(t,\tau) = \exp (-R_2^s (t-\tau)) \;\; \forall \;\; t \geq \tau,\]

for a Gaussian lineshape it is

\[ G(t,\tau) = \exp(- {R_2^s}^2 (t-\tau)^2 / 2)),\]

and for super-Lorentzian lineshape it is

\[ G(t,\tau) = \int_0^{1} \exp \left(- {R_2^s}^2 (t - \tau)^2 \cdot \frac{(3 \zeta^2 - 1)^2}{8} \right) d\zeta.\]

As evident from these equations, the Green's functions are merely a function of $\kappa = R_2^s \cdot (t - \tau) = (t - \tau) / T_2^s$, and in this package we implemented the functions as such: greens_lorentzian(κ), greens_gaussian(κ), and greens_superlorentzian(κ). These functions can be used to reproduce Fig. 1 in the generalized Bloch paper:

using MRIgeneralizedBloch
using Plots

T₂ˢ = 10e-6 # s
t = 0 : 1e-6 : 1e-3
p = plot(yaxis=:log, ylim=(1e-6,1), xlabel="(t-τ) [ms]", ylabel="G((t-τ)/T₂ˢ)")
plot!(p, 1e3t, greens_lorentzian.(t ./ T₂ˢ), label="Lorentzian lineshape")
plot!(p, 1e3t, greens_gaussian.(t ./ T₂ˢ), label="Gaussian lineshape")
plot!(p, 1e3t, greens_superlorentzian.(t ./ T₂ˢ), label="super-Lorentzian l.")

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