gftool.fourier.tt2z_trapz

gftool.fourier.tt2z_trapz(tt, gf_t, z)

Laplace transform of the real-time Green’s function gf_t.

Approximate the Laplace integral by trapezoidal rule:

\[G(z) = ∫dt G(t) \exp(izt) ≈ ∑_{k=1}^N [G(t_{k-1})\exp(izt_{k-1}) + G(t_k)\exp(izt_k)] Δt_k/2\]

The function can handle any input discretization tt.

Parameters

tt(Nt) float np.ndarray

The points for which the Green’s function gf_t is given.

gf_t(…, Nt) complex np.ndarray

Green’s function and time points tt.

z(…, Nz) complex np.ndarray

Frequency points for which the Laplace transformed Green’s function should be evaluated.

Returns

gf_z(…, Nz) complex np.ndarray

Laplace transformed Green’s function for complex frequencies z.

See also

tt2z_lin

Laplace integration using Filon’s method

Notes

The function is equivalent to the one-liner np.trapz(np.exp(1j*z[:, None]*tt)*gf_t, x=tt). Internally this function evaluates the sum as a matrix product to leverage the speed-up of BLAS. If numexpr is available, it is used for the speed up it provides for transcendental equations.