XSup

G pαp α ∈N∗
Z(G)6={1}
√2 + 3 √3
n≥3ω1, . . . , ωnn ωn= 1
p∈Z
Sp=
n
X
i=1
ωp
i
T=
n−1
X
i=1
1
1−ωi
z7→ 1
1−z
BC
z∈B1−z+z2∈B1 + z+z2∈B
B
(zn)n≥0z0∈C
∀n∈N, zn+1 =zn+|zn|
2
z∈C|z|<1
lim
n→+∞
n
Y
k=0 1 + z2k
(an) (un)
u0>0un+1 =1
un+an+ 1 n∈N
(un) (an)
α1/e
f∈ C1(R,R)
∀x∈R, f0(x) = αf(x+ 1)
α= 1/e
z∈C7→ zexp(z)∈C
f∈ C2(R+,R) limx→+∞f(x) = a∈R
f00 f0(x)x→+∞
f: [0 ; 1[ →Rf
(a, b)∈R2a < b f ∈ C0([a;b],C)
f
Zb
a
f =Zb
a|f|

(a, b)∈R2µ∈R∗
+f∈ C2([a;b],R)
∀x∈[a;b],f0(x)≥µ f0
Zb
a
e2iπf(t)dt≤1
µπ
n→+∞
In=Z1
0t
1 + t2n
dt
f: [0 ; 1] →R
Z1
0
f(t) dt= 0
x∈]0 ; 1[
Zx
0
tf(t) dt= 0
f: [0 ; 1] →R
Z1
0
f(t) dt= 0
m f M
Z1
0
f2(t) dt≤ −mM
n≥2
Pn=Xn−nX + 1
Pnxn
xnn→+∞
(xn)
xn
a, b ∈N\ {0,1}n∈N∗
an+bnn
P∈C[X]P(0) = 1
∀ε > 0,∃z∈C,|z|< ε P(z)<1
t∈Rn∈N∗
R[X] (Xcos t+ sin t)n
X2+ 1
P∈R[X]RP
P∈C[X]
P(X2) = P(X)P(X+ 1)
P∈R[X]Rα P 0+αP
R

P∈C[X]
P(1) = 1, P (2) = 2, P 0(1) = 3, P 0(2) = 4, P 00(1) = 5 P00(2) = 6
C3
x2+y2+z2= 0
x4+y4+z4= 0
x5+y5+z5= 0
Ka1, a2, . . . , an∈K
n
X
i=1 Y
j6=i
X−aj
ai−aj
A(X) = Qn
j=1 (X−aj)
n
X
i=1
1
A0(ai)
(a, b, c)∈R3x7→ sin(x+a), x 7→ sin(x+b)x7→ sin(x+c)
P∈R[X] (P(n))n∈N
E F1F2E
F1F2
Mn(R)
A, B ∈ Mn(R)
(AB −BA)2= In
M=a b
c d∈ M2(R)
0≤d≤c≤b≤a b +c≤a+d
n≥2
Mn=anbn
cndn
n≥2
bn+cn≤an+dn
A=
1··· ··· 1
0 1
0··· 0 1
∈ Mn(R)
k∈N∗Ak
A−1
(A−1)kk∈N

A1, . . . , Ak∈ Mn(R)
A1+··· +Ak= In∀1≤i≤k, A2
i=Ai
∀1≤i6=j≤k, AiAj= On
A∈ Mn(R)X, Y Mn,1(R)
A+YtX∈GLn(R)⇐⇒ 1 + tXA−1Y6= 0
Dn+1 =
C0
0C1 1··· Cn
n
C0
1C1 2··· Cn n+1
C0
nC1
n+1 ··· Cn 2n
[n+1]
Ck
n=n
k=n! k!(n−k)!
Pn(X) = Xn−X+ 1 n≥2
Pnn z1, . . . , znC
1 + z11··· 1
1 1 + z2
1
1··· 1 1 + zn
A H Mn(R) rg H= 1
det(A+H) det(A−H)≤det A2
(un)p∈N∗upn −un→0
(un)
n→+∞n
X
k=1k
nn
(un)
n
X
k=1
uk−nun
un
(un)n≥1
n∈N∗
vn=un/SnSn=u1+··· +un
Pvn
(un)
un
n
Rn=
+∞
X
k=n+1
uk
un/Rnun/Rn−1

RG
y1Ry2⇐⇒ ∃x∈G, xy1=y2x
RG
RG
GR
y
∀x∈G, xy =yx
y∈Z(G)
GR
Card G= Card Z(G) + N
NR
RG
{y1, . . . , yn} R
Hi={x∈G|xy1=yix}
i∈ {1, . . . , n}y1Ryixi∈G
xiy1=yixi
ϕ:H1→Hi
ϕ(x) = xix
Card H1=. . . = Card Hn=m
G H1, . . . , Hn
Card G=mn =pα
p p |N
pCard G= Card Z(G) + N
p|Card Z(G)
Z(G)6=∅1∈Z(G)
Card Z(G)≥p
x=√2 + 3
√3
x−√23= 3
x3−3√2x2+ 6x−2√2=3
3x2+ 2√2 = x3+ 6x−3√2 = x3+ 6x−3
3x2+ 2
x√2
Q
∀k∈ {1, . . . , n}, ωk= e2ikπ/n =ωkω= e2iπ/n
n p ωp6= 1
Sp=
n
X
k=1
ωkp =ωp1−ωnp
1−ωp= 0
n p
Sp=
n
X
k=1
ωkp =
n
X
k=1
1 = n
(a−b)3=a3 −3a2b+ 3ab2 −b3