inc_gamma_imag: Incomplete gamma function of imaginary argument

In fcheysson/hawkesbow: Estimation of Hawkes Processes from Binned Observations

Incomplete gamma function of imaginary argument

Description

Calculates the value of

\Gamma_1(x, \alpha) = \int_x^\infty t^{\alpha-1} e^{-it} \mathrm{d}t

for 0 < \alpha < 1 through the following relations:

\int_0^\infty t^{\alpha-1} e^{-it} \mathrm{d}t = e^{-i\frac{\pi}{2}\alpha} \int_0^\infty t^{\alpha-1} e^{-t} \mathrm{d}t = e^{-i\frac{\pi}{2}\alpha} \Gamma(\alpha).

obtained by contour integration, and:

\int_0^x t^{\alpha-1} e^{-it} \mathrm{d}t = \int_0^x t^{\alpha-1} \mathrm{cos}(t) \mathrm{d}t - i \int_0^x t^{\alpha-1} \mathrm{sin}(t) \mathrm{d}t = Ci(x, \alpha) - i Si(x, \alpha)

. The first integral is calculated using function "tgamma" from the library "boost::math", while the functions Ci and Si are approximated via Taylor expansions.

Usage

inc_gamma_imag(x, alpha)

Arguments

x	A non-negative number
alpha	A number between 0 and 1 (strictly)

Value

The incomplete gamma function of imaginary argument (see Details)

Examples

inc_gamma_imag(1.0, 0.5)

fcheysson/hawkesbow documentation built on Jan. 26, 2024, 9:30 p.m.