HP71B IBOUND fooled

[Albert Chan]

(05-21-2021 07:32 PM)Albert Chan Wrote:  F(∞) - F(0)
= re(cis(-1/4) * sqrt(pi)/2 * erf(w/2)/w)
≈ cos(1/4) * 0.496884029215 + sin(1/4) * 0.0414810242685
≈ 0.491699677694

Just found a more elegant way, without doing even cos(1/4), sin(1/4).

Abramowitz and Stegun, eqn 7.1.6: \(\quad\displaystyle erf(z) =
{2\over\sqrt{\pi}} e^{-z^2} \sum_{n=0}^∞ {2^n \over 1·3· \cdots (2n+1)} z^{2n+1} \)

exp(-z^2) = exp(-w^2/4) = exp(i/4) = cis(1/4)
This got completely cancelled with term cis(-1/4)

This left only summation terms ... and only the real terms

F(∞) - F(0)
= 1/2 - 1/(3*5*2^3) + 1/(3*5*7*9*2^5) - 1/(3*5*7*9*11*13*2^7) + ...
= 1/2 * (1 - 1/(3*5*4) * (1 - 1/(7*9*4) * (1 - 1/(11*13*4) * (1 - ...))))
≈ 0.491699677694