Euler representation of a complex number z = α + βi :
z = r eϕi,
where r = |z| and ϕ ∈ [0 , 2π[
(corresponds to polar coordinates of points in the euclidean plane).
Im
6
1
sin ϕ
s eϕi
s r eϕi
ϕ
cos ϕ
−1
1
(here: r > 1)
- Re
−1
70
Note: A geometric interpretation of complex multiplication:
For z1, z2 ∈ C, write z1 = r1eϕ1i and z2 = r2eϕ2i , then:
z1z2 = r1r2 e(ϕ1+ϕ2)i.
Complex representations of sine and cosine
cos ϕ = Re eϕi = 21 eϕi + e−ϕi ,
1
ϕi
ϕi
−ϕi
sin ϕ = Im e
= 2i e − e
.
Note: eϕi = cos(ϕ) − sin(ϕ)i = cos(−ϕ) + sin(−ϕ)i =
e−ϕi .
Example: Deriving trigonometric formula
cos(α + β) = cos(α) cos(β) − sin(α) sin(β):
l.h.s. = Re e(α+β)i = Re eαieβi
= Re cos(α) + sin(α)i cos(β) + sin(β)i
= cos(α) cos(β) − sin(α) sin(β) = r.h.s.
Example: One-dimensional harmonic oscillation
s(t) = a cos ωt + ϕ0 , t a real variable (time),
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a > 0 amplitude, ω > 0 angular frequency, ϕ0 ∈ [0 , 2π[ phase shift.
S(t) = A eωti ,
Complex representation:
amplitude.
Note: s(t) = Re S(t)
4.5
where A = aeϕ0i the complex
ϕ0 i ωti
e
since a cos(ωt+ϕ0) = Re ae(ωt+ϕ0)i = Re |ae{z
}
=A
Complex Polynomials
Well-known: A real polynomial of degree n
P (x) = cnxn + cn−1 xn−1 + . . . + c1x + c0 ,
(with real coefficients c0 , c1 , . . . , cn , where cn 6= 0)
x ∈ R,
may not have any zero, i.e., there may not exist an x0 ∈ R with P (x0 ) = 0 .
Example: A second degree polynomial P (x) = x2 + px + q , where p, q ∈ R
2
are such that ∆ = p2 − q < 0 , does not have any (real) zeros.
However, for a complex polynomial of degree n ,
P (z) = cnz n + cn−1 z n−1 + . . . + c1z + c0 , z ∈ C ,
(with complex coefficients c0 , c1 , . . . , cn , where cn 6= 0)
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the situation is completely different.
Fundamental theorem on complex polynomials
If P (z) is a complex polynomial of degree n ≥ 1,
then there exists a zero of P (z), i.e., a z0 ∈ C
such that P (z0) = 0 ;
and, moreover:
There exist z1, z2, . . . , zn ∈ C such that P (z) factorizes to
P (z) = cn(z − z1) (z − z2) · · · (z − zn) , for
all z ∈ C ,
(where cn 6= 0 is the highest coefficient of P (z) ).
Note: The complex numbers z1, z2, . . . , zn (the zeros of P (z)) need
not be all distinct.
Example: The above 2nd degree polynomial, but now with a complex variable:
P (z) = z 2 + pz + q , z ∈ C ,
73
2
where p, q ∈ R such that ∆ = p2 − q < 0 .
Not difficult to check:
p
p
P (z) = (z − z1) (z − z2) , where z1,2 = − 2 ± |∆| i .
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5
5.1
Derivatives of functions of one real variable
Basics on derivatives
Derivative of f at x0 :
f (x1 ) − f (x0 )
f (x0 + h) − f (x0 )
= lim
x1 →x0
h→0
x1 − x0
h
f ′(x0) = lim
Other symbols for the derivative f ′(x0) :
df (x) df
(x0 ) or
, etc.
dx
dx x=x0
Local linear approximation of f (x) :
The tangent as a (linear) function approximates f (x) if x is close to x0 :
f (x) ≈ f (x0 ) + f ′(x0) (x − x0).
f is called differentiable on its domain (usually an interval I ⊆ R), if the derivative f ′(x0) exists for each x0 ∈ I . Then x = x0 ∈ I is a variable, and we have
the function f ′(x) or shortly f ′ , the derivative of f . The derivative of f ′ at x0
75
is called the second derivative of f at x0 and is denoted by f ′′(x0) .
If f ′ is differentiable on its domain I then we have the function f ′′(x) or shortly
f ′′ , the second derivative of f .
(if it exists)
Local quadratic approximation of f (x) : If x is close to x0 then
1
f (x) ≈ f (x0 ) + f ′(x0 ) (x − x0) + f ′′(x0) (x − x0)2.
2
In particular, when f ′(x0) is nearly zero, then the local quadratic approximation
is considerably better than the local linear approximation.
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function f
derivative f ′
domain I
c (constant)
0
R
xm , (m ∈ N)
m xm−1
R
xm , (m ∈ Z, m ≤ −1)
m xm−1
xα , (α ∈ R)
α xα−1
] 0 , ∞ [ or ] − ∞ , 0 [
ln x
ex
1
x
x
e
sin x
cos(x)
cos x
− sin(x)
tan x
] 0 , ∞[
R
1
cos2 (x)
cot x
− sin21(x)
arcsin x
√ 1
1−x2
1
− √1−x
2
1
1+x2
arccos x
arctan x
] 0 , ∞[
77
R
R
1
2 )π ,
1
2 )π
(k −
(k +
, (k ∈ Z)
kπ , (k + 1)π , (k ∈ Z)
] − 1, 1[
] − 1, 1[
R
Useful rules for derivatives
For functions f and g on a common domain I :
d αf (x) + βg(x) = αf ′(x) + βg ′(x) , (for α, β ∈ R) ;
dx
d f (x) g(x) = f ′(x) g(x) + f (x) g ′(x) ;
dx
d f (x)
f ′(x) g(x) − f (x) g ′(x)
, (if g(x) 6= 0 for all x ∈ I).
=
2
dx g(x)
g(x)
If ϕ = ϕ(x) and f = f (y) are two functions such that the composition f ϕ(x)
exists, then the chain rule says:
d
f ϕ(x) = f ′(y) ϕ′(x) with y = ϕ(x) .
dx
d
d λx
2
λx
2
Examples: dx e = λ e , dx exp −c x = −2c x exp −c x
h i
d
1
d 1
x
−x
x
−x
= 2 e −e
= sinh x ;
dx cosh x = dx 2 e + e
d
dx
x − cos(x) sin(x)
= 1 + sin2(x) − cos2(x) = 2 sin2(x) ;
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d x2 −1
dx x2 +1
d
dx
5.2
=
arctan
2x(x2 +1)−(x2 −1)2x
(x2 +1)2
√
x =
1
1
√
1+( x)2 2
=
4x
(x2 +1)2
1
x− 2 =
;
1 √
2(1+x) x
.
Derivatives of vector-valued functions
Vector-valued function of one real variable:
f (x) = f1(x) , f2(x) , . . . , fp(x) , x ∈ I ⊆ R , i.e., f (x) ∈ Rp. The fi are
called the component functions. Derivative of f at x0 ; derivatives f ′ and f ′′
are explained as follows:
f (x1) − f (x0 )
.
x1 →x0
x1 − x0
′
′
′
′
Easy to see: f (x0) = f1(x0) , f2(x0) , . . . , fp(x0) . If the derivatives exist
at each x0 = x ∈ I then the derivative f ′ of f is the Rp-valued function
′
′
′
′
f (x) = f1(x) , f2(x) , . . . , fp(x) , , x ∈ I.
f ′(x0 ) = lim
If, furthermore, the derivatives of f ′ exist on I, then we may also define a second
79