gftool.fourier.tt2z_lin
- gftool.fourier.tt2z_lin(tt, gf_t, z)[source]
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 at time points tt.
- z(…, Nz) complex np.ndarray
Frequency points for which the Laplace transformed Green’s function should be evaluated.
- Returns:
- (…, 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
If numexpr is available, it is used for the significant speed up it provides for transcendental equations. Internally the sum is evaluated as a matrix product to leverage the speed-up of BLAS.
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