The Evaluation of the integral ∫_r^∞(exp(-z)/z²)dz

The integral of ∫_r^∞(exp(-z)/z²)dz in important in the physics of atomic nuclei. For convenience define

w(r) = ∫_r^∞(exp(-z)/z²)dz
 

This function can be evaluated by numerical approximation over an inteval of r to R by dividing the interval into N increments of Δr and using the trapezoidal rule; i.e.,

w(r) ≅ Σ ½[f(r_i)+f(r_i+1)Δr
which reduces to
w(r) ≅ [½f(r₀) + Σf(r_i) + ½f(r_N)]Δr
 

where Δr=(R-r)/N and r_i=r+i*Δr.

The error involved in truncating the integration at R is bounded by exp(-R)/R. Thus if R=10 then the error due to the truncation is less than 4.54×10^-6 and if R=15 then the error is less than 2.04×10^-8. For R=20 the truncation error is less than 1.03×10^-10 In addition to the error from the truncation of the integral there is error from applying the trapezoidal rule to a function that is curved.

Using R=20 and the trapezoidal quadrature method some of the values of w(r) are