Test de primitives amstex

Part I
Some tests

1  Matrices

Toutes les références

Les numéros sont :

1.b3.a3.b3.c

1.1  Simple

10
01
  


10
01


  






10
01
12






    (1.a)

1.2  Compliqué



























1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20


























    (1.b)

2  Environement cases

To summarize, we obtain the following exact representation for the inverse of x2ex+1:







Y(x) = y(logx),  y0 inverse of 2log x+x+log(1+ex/x2),
y[x+log(1+y0−2(x)ey0(x))] = y0(x),  y0 inverse of x+2logx,
y0(x) = y1(logx),  y1 inverse of logx+log(1+2logx/x),
y1(x) = exp(y2(x)),  y2 inverse of x+log(1+2xex),
y2[x+log(1+2y3(x)ey3(x))] = y3(x),y3 inverse of x.

3  Test d’alignement

Now the φis are very easy to compute:

     
φ1  =  y0 = 1/t2,         
φ2
  = φ1(y0(x+g))−φ1 = 
t3t22
(1+2t2)
1+4t2+2t22
2(1+2t2)3
t24t32+O(t33),
         
φ3
  = φ2(y0(x+g))−φ2 = −
1+4t2+2t22
(1+2t2)3
t24t32+O(t33),
         

A similar treatment applies to (1.b), and leads to

     
x+log(1+2xex)  =  1/t1(y3(x+g))       =  x+2xex−2x2e−2x+O(x3e−3x),       
exp[−x+log(1+2xex)]  =  t2(y3(x+g))       =  ex−2xe−2x+4x2e−3x+O(x3e−4x).        

3.1  gather, multline

or one of the following (successive) refinements:

  exp(eU)


1−
2eU1/2
U1/2+4
+
2e−2U1/2
(U1/2+4)2
4
3
e−3U1/2
(U1/2+4)3
+O(e−4U1/2)


,
    (3.a)
exp(eU)exp


2eU1/2
U1/2+4






1+
8−2U−1/2U−1+U−3/2
(4+U−1/2)3
eU−2U1/2+O(e−2U)


, 
    (3.b)

Cette équation a le numéro 3.c, celles d’au dessus les numéro 3.a et 3.b.

  1/t2(y0(x+g)) = y0(x+log(1+y0−2ey0))
y0+
ey0
y02(1+2/y0)
1+4/y0+2/y02
2y04(1+2/y0)3
e−2y0+O(e−3y0).

    (3.c)

Part II
Other tests

4  Zéro

x2+y2 = z2     (4.a)
x2+y2 = z2
x2+y2 = z2     (4.b)
x2+y2 = z2
x2+y2 = z2     (oups)

Au desus 4.a, 4.b et oups. Après 4.b

5  Deux

x2+y2 = z2    (5.a)
x3+y3 = z3    (5.b)
x4+y4 = z4    (5.c)
x5+y5 = z5    (5.d)
x6+y6 = z6    (5.e)
x7+y7 = z7     (5.f)

6  Un

x2+y2 = z2      (6.a)
x3+y3 = z3 
x4+y4 = z4      (*)
x5+y5 = z5     *
x6+y6 = z6     6.a
x7+y7 = z7     (6.b)

Ben mon vieux pour avoir l’équation *.

7  Trois

x2+y2 = z2    (+)
x3+y3 = z3
x4+y4 = z4
x5+y5 = z5
x6+y6 = z6
x7+y7 = z7

8  Quatre

x2+y2 = z2    (8.a)
x3+y3 = z3    (8.b)
x4+y4 = z4    (8.c)
x5+y5 = z5    (8.d)
x6+y6 = z6    (8.e)
x7+y7 = z7     (8.f)

Maintenant : 8.f.

9  Cinq

     
 x2+y2z2     x3+y3z3           (a)
 x4+y4z4     x5+y5z5        
x6+y6z6     x7+y7z7           (9.a)

Au dessus ya l’équation (a).

10  Multline

x+1 = y+2

    (10.a)


This document was translated from LATEX by HEVEA.