<p> plot(data,a,b,c) </p>

63c7f16f-17a1-4878-8660-0e7c6b784cac 20 4 5 decimal

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data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

Laguerre polynomials

n x Laguerre 1 - (k^n! * x k^* (k! (2^n k - (! * / k 0 n sum :

x λ 10 x ≤ (x 32 ≤ (& cases :

j 04 range :

j x Laguerre vectorize 11 x - 2 x 2^+ 4 x * - 2 / 323 x 2^* + (* - x 3^+ 18 x * + 6 / - 832 x 3^* - (* x 4^+ 24 x * 4 - 3 x * + (* + 24 / 51 mat

1 x Laguerre x λ * 2 x Laguerre x λ * 3 x Laguerre x λ * 4 x Laguerre x λ * 5 x Laguerre x λ * 5 1 sys

n Ω n x Laguerre x coeffs maple : n 10 : c n Ω vectorize eval : r c polyroots : r r eval : 51 line

Wr i 1 n range W i el n! (2^n 1 + (! (2^/ r i el n 1 + r i el Laguerre (2^/ * : for r W W r exp * vectorize augment 21 line :

x - exp x f * x 0 ∞ int w i el x i el f * i 0 n sum ≡

11½  Decimals sanity Mathcad 11 ... Abramowitz-Stegun Wr 0.13779347054 0.308441115768 0.35400973861 0.729454549503 0.401119929155 0.831902301043 1.80834290174 0.218068287612 1.33028856175 3.401433697882 0.062087456088 1.863063902849 5.55249613842 0.009501517034 2.450255569293 8.330152778362 0.000753008329 3.122764005093 11.843785595256 0.000028259246 3.934153433468 16.279258664712 0.000000424931 4.992413665787 21.996584524241 0.00000000184 6.572202101694 29.920697766435 0.000000000001 9.784698040466 103 mat

x g x 0 ∞ int w i el x i el exp * x i el g * i 0 n sum ≡

Plot roots of Laguerre roots r r 0 * augment : roots roots . 14 black plot : 21 line

roots n x Laguerre x λ * 2 1 sys

Laguerre polyroots otherwise . . . from Andrey Ivashov

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data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

Степень полинома Pangkat polinomial Polynomial order for polyroots n 5 :

z n L e z^n! / e z -^z n^* (z n diff * :

T 1 el z n L : k 1 n range T k 1 + el T k el (z diff k / : for z D T : 21 line 21 line

Коэффициенты полинома Лагерра Koefisien polinomial Laguerre Laguerre polyroots coefficients v 0 D : 15 - 553 / - 524 / 1120 / - 61 mat

HermiteRoots v polyroots : r HermiteRoots eval : roots r r 0 * augment : roots roots . 14 black plot : 41 sys

x λ 10 x ≤ (x 20 ≤ (& cases :

plot roots x n L x λ * 21 sys :

plot

sanity Abramowitz 12 D r 0.263560319718 1.413403059107 3.596425771037 7.085810005862 12.640800844276 51 mat