where E[n,z] = [[integral].sup.[infinity].sub.1] [t.sup.-n] [e.sup.-zt] is 'exponential integral
Salem, "A q-analogue of the exponential integral
," Afrika Matematika, vol.
However, the incomplete gamma function is a more complete way to calculate the temperature integral than the exponential integral
because [p.sub.m](x) = [LAMBDA](-(m + 1),x) yielded consistent results (low errors) for any m, integer or non-integer in the interval 1 [less than or equal to] x [less than or equal to] 100 tested in the supplementary material.
Among the topics are complete elliptic integrals, the Riemann zeta function, some automatic proofs, the error function, hyper-geometric functions, Bessel-K functions, polylogarithm functions, evaluation by series, the exponential integral
, confluent hyper-geometric and Whittaker functions, and the evaluation of entries in Gradshteyn and Ryzhik employing the method of brackets.
where [[bar.[gamma]].sub.bm] = [P.sub.b]/[[sigma].sup.2.sub.m] and [E.sub.1] (x) = [[integral].sup.[infinity].sub.x]([e.sup.-t]/t)dt is the exponential integral
A.1 Mathematical tools: Exponential integral
, sine integral and cosine integral
For indefinite integrals, an extended version of the Risch code is used whenever both the integrand and integral can be expressed in terms of elementary functions, exponential integral
functions, polylogarithms and other related functions.
and ExpIntegral E[n, z] gives the exponential integral
function [E.sub.n] (z), which is defined by
, Logarithmic Integral, Sine and Cosine Integrals
In Section 2, we provide an explicit expression of the moment generating function in terms of the exponential integral
We define, for x > 0, the function F(x) = x[e.sup.x][E.sub.1](x), where [E.sub.1] (x) denotes the exponential integral
Uniform convergence and computation of the generalized exponential integrals