The above question is of course a bit too general, but basically it would be of great advantage to find a way to ensure that numerical integration of a fraction that has an exponential function in its numerator consistently produces reliable results.
According to Gradshteyn and Ryzhik, the following correlation exists for the exponential integral:
When I try to calculate the integral on the right hand side unsing MATLAB's numerical integration, and check the result by comparing it to the analytical formula (i.e. exp(-x)*ei(x)-1/x obtained from the above equation) the result is very noisy. Here is the script:
clc; format long g; clear all; close all;
fun = @(t) exp(-t)./(x(ii)-t).^2;
y(ii) = integral(fun,0,Inf,'RelTol',1e-8);
Y(ii) = exp(-x(ii))*ei(x(ii))-1/x(ii);
plot(x,y,'k-',x,Y,'r-','linewidth',1.5)
Y is the analytical calculation and y is the result of direct numerical integration of the function exp(-t)/(x-2)^2 between 0 and infinity.
Is there a way to change MATLAB's numerical integration parameters to improve the quality of this group of integrals?
