1 Rappel de quelques commandes utiles

f2(x) = x−x3
3!
f2 := x−>(x−x3/3!);
I=R1
0f2(x)dx
float(int(f2(x),x=0..1));
f2[0; 1]
f2
f2(x) = x−x3
3!
DEFINE(F2(X) = X−X3/3!)
I=R1
0f2(x)dx
R1
0F2(X)dX
f(x) = x x < π f(x) = 2π−x x ≥π
DEFINE(F(X) = IFTE(X< π, X,2π−X))
f2(x) = x−x3
3!
Define f2(X) = X−X3/3!
I=R1
0f2(x)dx R(f2(X),X,0,1)
ENTER
f(x) = x x < π f(x) = 2π−x x ≥π
Define f(X) = when(X< π, X,2π−X)

y= sin(x)y=x
y= sin(x)
y=x
f(x)n
cnf(x)
n
cn
f]−α;α[
x= 0 ann∈NR≤α
f(x) =
∞
X
n=0
anxn|x|< R (R > 0)
f an
an=f(n)(0)
n!
x=x0f(x)ann∈N
R
f(x) =
∞
X
n=0
an(x−x0)n|x−x0|< R

x= 0 sin(x)
S(x)x= 0 sin(x)
f(x) = sin(x)
f1(x) = x
f2(x) = x−x3
3!
f3(x) = x−x3
3! +x5 5!
f4(x) = x−x3
3! +x5 5! −x7 7!
f4(x) sin(x)
R4(x)S(x)
f4(x) sin(x)x∈[−1,1]
sin(1)
[0; π] sin(x) 10−6
x= 0 k
fk(x) =
k
X
j=1
(−1)j−1x2j−1
(2j−1)!
sin(3) sin(π−3)
π10−10
I=Zπ
0
sin(x)
xdx
x= 0
g(x) = sin(x)
x
I vj
RnRn=
∞
X
j=n+1
vj|Rn|
I
I, Pk
j=0 vjk

I=Z1
2
0
1
√1 + x3dx
x= 0
g(x) = 1
√1 + x3
I vn
I10−6
x= 0
θ(x) = 1
√πZx
0
e−t2dt
θ(1) 10−3
2π
n∈Zα∈R
cn(f) = 1
2πZα+2π
α
f(t)e−intdt
f
SF (f)(x) =
+∞
X
n=−∞
cn(f)einx
n∈Nα∈R
an(f) = 1
πZα+2π
α
f(t) cos(nt)dt
bn(f) = 1
πZα+2π
α
f(t) sin(nt)dt
SF (f)(x) = a0(f)
2+
+∞
X
n=1
(an(f) cos(nx) + bn(f) sin(nx))
x0f
f(x0+ 0) f(x0−0)
SF (f)(x0)1
2(f(x0−0) + f(x0+ 0))

f x SF (f)(x)f(x)
SF (f)(x) =
∞
X
k=0
ak(f) cos(kx)+bk(f) sin(kx)
f2π
f(x) = x]−π;π[
f(π) = 0
SF (f)n(x) =
n
X
k=0
ak(f) cos(kx) + bk(f) sin(kx)
x∈[−4; 4]
f(x)
SF (f)1(x) =
SF (f)2(x) =
SF (f)3(x) =
SF (f)4(x) =
SF (f)5(x) =
SF (f)6(x) =
SF (f)n
xn, yn
f n +∞
xnπ
ynα= 2 Zπ
0
sin(t)
tdt
α α > 3.7> π
yn=SF (f)n(xn)n+∞
xn, yn
SF (f)n(x) = Pn
k=0 ak(f) cos(kx) + bk(f) sin(kx)
xn
SF (f)n(x) = 2 Pn
k=1 (−1)k+1 sin(kx)
k