E1_imaginary

Calculates the value of
$$E_1(ix) = \int_1^\infty \frac{e^{-ixt}}{t} \mathrm{d}t$$
using its relation to the trigonometric integrals
(cf. <a href="https://en.wikipedia.org/wiki/Exponential_integral#Exponential_integral_of_imaginary_argument">https://en.wikipedia.org/wiki/Exponential_integral#Exponential_integral_of_imaginary_argument</a>):
$$E_1(ix) = i \left[ -\frac{1}{2} \pi + Si(x) \right] - Ci(x)$$
and their Pad\'e approximants
(cf. <a href="https://en.wikipedia.org/wiki/Trigonometric_integral#Efficient_evaluation">https://en.wikipedia.org/wiki/Trigonometric_integral#Efficient_evaluation</a>)

Implements an estimation method for Hawkes processes when count data are only observed in discrete time, using a spectral approach derived from the Bartlett spectrum, see Cheysson and Lang (2020) <arXiv:2003.04314>. Some general use functions for Hawkes processes are also included: simulation of (in)homogeneous Hawkes process, maximum likelihood estimation, residual analysis, etc.

Felix Cheysson

hawkesbow

Estimation of Hawkes Processes from Binned Observations

E1_imaginary function

<dl><dt>x</dt>
<dd>A non-negative number</dd></dl>

Arguments

Calculates the value of
$$E_1(ix) = \int_1^\infty \frac{e^{-ixt}}{t} \mathrm{d}t$$
using its relation to the trigonometric integrals
(cf. <a href='https://en.wikipedia.org/wiki/Exponential_integral#Exponential_integral_of_imaginary_argument'>https://en.wikipedia.org/wiki/Exponential_integral#Exponential_integral_of_imaginary_argument</a>):
$$E_1(ix) = i \left[ -\frac{1}{2} \pi + Si(x) \right] - Ci(x)$$
and their Pad\'e approximants
(cf. <a href='https://en.wikipedia.org/wiki/Trigonometric_integral#Efficient_evaluation'>https://en.wikipedia.org/wiki/Trigonometric_integral#Efficient_evaluation</a>)

Exponential integral of imaginary argument — E1_imaginary

<dl>

<dt>x</dt>
<dd>A non-negative number</dd>

</dl>

E1_imaginary: Exponential integral of imaginary argument

Description

Usage

Value

Arguments

Examples