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 [filon1930] and [iserles2006].

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

filon1930

Filon, L. N. G. III.—On a Quadrature Formula for Trigonometric Integrals. Proc. Roy. Soc. Edinburgh 49, 38–47 (1930). https://doi.org/10.1017/S0370164600026262

iserles2006

Iserles, A., Nørsett, S. P. & Olver, S. Highly Oscillatory Quadrature: The Story so Far. in Numerical Mathematics and Advanced Applications (eds. de Castro, A. B., Gómez, D., Quintela, P. & Salgado, P.) 97–118 (Springer, 2006). https://doi.org/10.1007/978-3-540-34288-5_6 http://www.sam.math.ethz.ch/~hiptmair/Seminars/OSCINT/INO06.pdf