gftool.fourier.tt2z_lin

gftool.fourier.tt2z_lin(tt, gf_t, z)

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

Filon’s method is used to calculate the Laplace integral

\[G(z) = ∫dt G(t) \exp(izt),\]

\(G(t)\) is approximated by a linear spline. The function currently requires an equidistant tt. Information on oscillatory integrations can be found e.g. in [filon1928] and [iserles].

Parameters:

tt : (Nt) float np.ndarray

The equidistant 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.

Raises:

ValueError

If the time points tt are not equidistant.

See also

tt2z_trapz

Plain implementation using trapezoidal rule.

Notes

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.

References

[filon1928]

L.N. Filon, On a quadrature formula for trigonometric integrals, Proc. Roy. Soc. Edinburgh 49 (1928) 38-47.

[iserles]

A. Iserles, S.P. Nørsett, and S. Olver, Highly oscillatory quadrature: The story so far, http://www.sam.math.ethz.ch/~hiptmair/Seminars/OSCINT/INO06.pdf