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In an old journal article on hydrology (CV Theis, Transactions - American Geophysical Union, Vol 22 part 3 pp 734 - 738, 1941), the following integral is presented:
P = (2/pi) * (integral from 0 to infinity) of {[e ^ (-k (1 + z^2))]/(1 + z^2)} dz
It is approximated as a series = 1 - e^(-k/2) (1.273239k + 0.424413k^2 + 0.183912k^3 + 0.038399k^4 + 0.008859k^5 ...)^(1/2).
It states "this series converges rapidly for fractional values of k. For larger values of k the integral can be more easily evaluated graphically or by Simpson's rule in the following form...."
P = (2/pi) * (integral from zero to pi/2) of e^(-k sec^2 u) du
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