Etheta_imaginary {hawkesbow}

Incomplete gamma function of imaginary argument with arbitrary power

Description

Calculates the value of

-ix e^{ix} E_\theta(ix) = -ix e{ix} \int_1^\infty t^{-\theta} e^{-ixt} \mathrm d t

for \theta > 0. This is achieved using recursive integrations by parts until 0 < \theta \le 1, then using either the exponential integral E1_imaginary if \theta = 1, or the incomplete gamma function inc_gamma_imag if 0 < \theta < 1.

Usage

Etheta_imaginary(theta, x)

Arguments

theta

A strictly positive number

x

A vector of non-negative numbers

Value

The incomplete gamma function of imaginary argument with arbitrary power (see Details)

Examples

Etheta_imaginary(3.14, 1.0)