SAN JOSÉ STATE UNIVERSITY

applet-magic.com
Thayer Watkins
Silicon Valley
USA

## 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(ri)+f(ri+1)Δr which reduces to w(r) ≅ [½f(r0) + Σf(ri) + ½f(rN)]Δr

where Δr=(R-r)/N and ri=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

rw(r)
6 0.000053193
5 0.000200001
4 0.000803216
3 0.003569475
2 0.01894927
1 0.151783879
0.5 0.690505202
0.2 3.507477572
0.1 11.59500065
0.05 41.61679933

A Calculator for the
w(r) Function
Variable r
Variable R