**SAN JOSÉ STATE UNIVERSITY****
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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(r_{0}) + 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

r | w(r) |
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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 |