fa2c4e05-243f-4c09-a234-20f8bbf59861 9 4 5 decimal

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"RootDichotomy"  is a robust companion. # Φ a b ε D RootDichotomy xn b a + (2 / : n 0 : xn Φ abs ε ≥ xn Φ a Φ * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while xn 51 line :

"RootSecant" is viable  => guess wisely tial/error Φ a b ε X Root Secant us 1 el ε : us 2 el b b Φ a b - (* a Φ b Φ - / - : i 2 : us i 1 + el us i el us i el Φ us i 1 - el us i el - (* us i 1 - el Φ us i el Φ - / - : us i el us i 1 - el - abs ε ≥ us i 1 + el us i el us i el Φ us i 1 - el us i el - (* us i 1 - el Φ us i el Φ - / - : i i 1 + : 21 line while collect last 'us'... index 1 explode us us rows row 1 el 81 line :

BesselJ(x,0)... is strctly 0 from x=0 ... ∞ Thus, you can multiply by anything. As a desirable effect, it constrains x*BesselJ(x,1)to start @ 0 for profitable inspection suite. Psi(x)+BesselJ(x,0)renders superbly Psi(x).

x f x x 1 BesselJ * π x 0 BesselJ * + :

h 0.001 :

x D x 2 h * + f - 8 x h + f * + 8 x h - f * - x 2 h * - f + 12 h * / :

0.5 D 0.4673

x f x D 0.5879 1 0.5879 0.5 - 2 2 mat 3 1 sys

x Γ x Gamma :

n x J 1 - (k^k! (k n + 1 + Γ * / x 2 / (n 2 k * +^* (k 011 sum :

0 x J 1 x J 2 1 sys

x Φ x f x D - (:

ε 106 -^:

above max Φ 0.5 0.75 ε X #

x 0.58790918391704 :

x_not defned x f 0.2 - x 04 solve #

x Φ 2.8 1012 -^* -

x Psi x 0 BesselJ + x ln 2 1 sys