MATHEMATICAL ANALYSIS I
Summary - v1.0.2
Basri Kerem Alhan
Selen Çelik
Zeynep Bıçakçıoğlu
Politecnico di Torino
,Chapter 1 - Sets of Numbers:
1.6: A subset A of R is called bounded from above or upper bounded if there exists a
real number b such that
x ≤ b, for all x ∈ A .
Applies for lower bound as well.
1.9: Let A ⊂ R be bounded from above. The supremum or least upper bound of A is
the smallest of all upper bounds of A, denoted by supA. The number s = supA is
characterised by two conditions:
i) x ≤ s for all x ∈ A;
ii) for any real r < s, there exists x ∈ A such that x > r.
Applies for infimum (greatest upper bound) as well.
Remark: Supremum may exists and not be a maximum, but when a maximum exists,
then it is also the supremum of the set.
(k)
n! n n(n − 1) . . . (n − k + 1)
1.11: Permutation, = Combination.
k!(n − k)! k!
Chapter 2 - Functions:
{−x i f x < 0;
x i f x ≥ 0,
2.1: The absolute value function: f : R → R, f (x) = | x | =
+1 i f x > 0,
The sign function: f : R → Z, f (x) = sign(x) = 0 i f x = 0,
−1 i f x < 0;
Floor function (Integer part): f : R → Z, f (x) = [x] = the greatest integer ≤ x
The mantissa: f : R → R, f (x) = M(x) = x − [x]
2.3: A map with values in Y is called onto if im f = Y. This means that each y ∈ Y is the
image of one element x ∈ X at least. The term surjective has the same meaning.
A function is called one-to-one or injective if every y ∈ im f is the image of a unique
element x ∈ dom f.
A function is invertible if it is bijective.
2.13: Even function (with respect to the y axis) if: f (−x) = f (x).
Odd function (with respect to the origin) if: f (−x) = −f (x).
, Chapter 3 - Vectors and Complex Numbers:
3.1: Polar Coordinates rcosθ, y = rsinθ
y
arctan x , i f . x > 0,
y
arctan x + π, i f x < 0, y ≥ 0,
y
3.2: r = x2 + y2, θ = arctan x − π, i f x < 0, y < 0
π
2
, i f x = 0, y > 0,
−π
2
, i f x = 0, y < 0.
3.3: The sum of vectors: v + w = (v1 + w1, . . . , vd + wd ) .
3.4: The product of vectors: λv = (λv1, . . . , λvd )
3.5: The Euclidean norm, or length, of a vector v with end-point P is defined:
d v12 + v22 i f d = 2,
vi2 =
∑
||v|| =
i=1 v12 + v22 + v32 i f d = 3.
3.26: Real and Imaginary part of z:
z + z̃ z − z̃
ℜez = , ℑm z =
2 2i
3.30: Exponential form or Euler formula:
e θi = cosθ + isinθ.
3.31: Exponential form of z:
z = re iθ.
3.38: Additional forms:
e z = e x e iy = e x(cos y + isiny) .
Summary - v1.0.2
Basri Kerem Alhan
Selen Çelik
Zeynep Bıçakçıoğlu
Politecnico di Torino
,Chapter 1 - Sets of Numbers:
1.6: A subset A of R is called bounded from above or upper bounded if there exists a
real number b such that
x ≤ b, for all x ∈ A .
Applies for lower bound as well.
1.9: Let A ⊂ R be bounded from above. The supremum or least upper bound of A is
the smallest of all upper bounds of A, denoted by supA. The number s = supA is
characterised by two conditions:
i) x ≤ s for all x ∈ A;
ii) for any real r < s, there exists x ∈ A such that x > r.
Applies for infimum (greatest upper bound) as well.
Remark: Supremum may exists and not be a maximum, but when a maximum exists,
then it is also the supremum of the set.
(k)
n! n n(n − 1) . . . (n − k + 1)
1.11: Permutation, = Combination.
k!(n − k)! k!
Chapter 2 - Functions:
{−x i f x < 0;
x i f x ≥ 0,
2.1: The absolute value function: f : R → R, f (x) = | x | =
+1 i f x > 0,
The sign function: f : R → Z, f (x) = sign(x) = 0 i f x = 0,
−1 i f x < 0;
Floor function (Integer part): f : R → Z, f (x) = [x] = the greatest integer ≤ x
The mantissa: f : R → R, f (x) = M(x) = x − [x]
2.3: A map with values in Y is called onto if im f = Y. This means that each y ∈ Y is the
image of one element x ∈ X at least. The term surjective has the same meaning.
A function is called one-to-one or injective if every y ∈ im f is the image of a unique
element x ∈ dom f.
A function is invertible if it is bijective.
2.13: Even function (with respect to the y axis) if: f (−x) = f (x).
Odd function (with respect to the origin) if: f (−x) = −f (x).
, Chapter 3 - Vectors and Complex Numbers:
3.1: Polar Coordinates rcosθ, y = rsinθ
y
arctan x , i f . x > 0,
y
arctan x + π, i f x < 0, y ≥ 0,
y
3.2: r = x2 + y2, θ = arctan x − π, i f x < 0, y < 0
π
2
, i f x = 0, y > 0,
−π
2
, i f x = 0, y < 0.
3.3: The sum of vectors: v + w = (v1 + w1, . . . , vd + wd ) .
3.4: The product of vectors: λv = (λv1, . . . , λvd )
3.5: The Euclidean norm, or length, of a vector v with end-point P is defined:
d v12 + v22 i f d = 2,
vi2 =
∑
||v|| =
i=1 v12 + v22 + v32 i f d = 3.
3.26: Real and Imaginary part of z:
z + z̃ z − z̃
ℜez = , ℑm z =
2 2i
3.30: Exponential form or Euler formula:
e θi = cosθ + isinθ.
3.31: Exponential form of z:
z = re iθ.
3.38: Additional forms:
e z = e x e iy = e x(cos y + isiny) .
