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Week 1

Complex numbers: the basics

1. The definition of complex numbers and basic operations

2. Roots, exponential function, and logarithm

3. Multivalued functions, or dependences

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,i yxz

where i ‘marks’ the second component. x and y are called the real and imaginary parts of z and are denoted

1. The definition of complex numbers and basic operations

Example 1:

۞ The set of complex numbers can be viewed as the Euclidean vector space R2, of ordered pairs of real numbers (x, y), written as

Like any vectors, complex numbers can be added and multiplied by scalars.

.Im,Re zyzx

Calculate: (a) (1 + 3i) + (2 – 7i) , (b) (–2)×(2 – 7i).

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22 yxz

۞ Given a complex number z =x + i y, the +tive real expression

is called the absolute value, or modulus of z. It’s similar to the absolute value (modulus, norm, length) of a Euclidean vector.Theorem 1: polar representation of complex numbers

A complex number z = x + i y can be represented in the form

,sinicos rz

where r = | z | and θ is the argument of z, or arg z, defined by

.cos,sinr

y

r

x

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Comment:

arg z is measured in radians, not degrees! You can still use degrees for geometric illustrations.

Like any 2D Euclidean vectors, complex numbers are in a 1-to-1 correspondence with points of a plane (called, in this case, the complex plane).

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Example 2:

Show on the plane of complex z the sets of points such that:

(a) | z | = 2, (b) arg z = π/3.

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Example 3:

Show z = 1 + i on the complex plane and find θ = arg z. How many values of θ can you come up with?

Thus, arg z is not a function, but a multivalued function, or a dependence.

۞ The principal value of the argument of a complex number z is denoted by Arg z (with a capital “A”), and is defined by

Multivalued functions will be discussed in detail later. In the meantime, we introduce a single-valued version of arg z.

.Arg,)(Argcos,)(Argsin zr

yz

r

xz

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Comment:

The graph of arg z looks like a spiral staircase.

Arg z

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For any z1 and z2, it holds that | z1 + z2 | ≤ | z1 | + | z2 |.

Theorem 2: the Triangle Inequality

,2121 zzzz

Proof (by contradiction):

Assume that Theorem 2 doesn’t hold, i.e.

hence...

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Our strategy: get rid of the square roots – cancel as many terms as possible – hope you’ll end up with something clearly incorrect (hence, contradiction).

.)()( 22

22

21

21

221

221 yxyxyyxx

The l.h.s. and r.h.s. of this inequality can be assumed +tive (why?) – hence,

This inequality is clearly incorrect (why?) – hence, contradiction. █

Since the l.h.s. and r.h.s. of (1) are both +tive (why do we need this?), we can ‘square’ them and after some algebra obtain

.)()( 22

22

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212121 yxyxyyxx

.2 21

22

22

212121 yxyxyyxx

(1)

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.)(i 1221212121 yxyxyyxxzz

۞ The product of z1 = x1 + i y1 and z2 = x2 + i y2 is given by

Example 4:

In addition to the standard vector operations (addition and multiplication by a scalar), complex numbers can be multiplied, divided, and conjugated.

Calculate (1 + 3i) (2 – 7i).

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,1i

or

Example 5:

,1)i10()i10(ii

Observe that

.1i2

One cannot, however, write

because the square root is a multivalued function (more details to follow).

Remark:

When multiplying a number by itself, one can write z×z = z2, z×z×z = z3, etc.

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Example 6:

.12 zzz

۞ The quotient z =z1/z2, where z2 ≠ 0, is a complex number such that

Calculate (1 + 3i)/(2 – 7i).

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Theorem 3: multiplication of complex numbers in polar form

.)](sini)([cos 21212121 rrzz

Useful formulae:

sin θ1 cos θ2 + sin θ2 cos θ1 = sin (θ1 + θ2),

cos θ1 cos θ2 – sin θ1 sin θ2 = cos (θ1 + θ2).

Proof: by direct calculation.

where r1,2 = | z1,2 | and θ1,2 = Arg z1,2.

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Theorem 5: The de Moivre formula

.sinicos)sini(cos nnn

This theorem follows from Theorem 4 with | z | = 1.

Theorem 4:

,)sini(cos nnrz nn

where r = | z | and θ = Arg z.

This theorem follows from Theorem 3.

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,2112 zzzz

۞ Complex numbers z1 = x + i y and z2 = x − i y are called complex conjugated (to each other) and are denoted by

.** 2112 zzzz or

Example 7:

If z = 5 + 2i, then z* = 5 – 2i.

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Theorem 6:

.2

zzz

Proof: by direct calculation.

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2. Roots, the exponential function, and the logarithm

۞ The nth root of a complex number z is a complex number w such that

.zwn

The solutions of equations (2) are denoted by

(2)

.n zw

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where r = | z |, θ = Arg z, and k = 0, 1... n – 1.

Theorem 7:

,2

sini2

cos

n

k

n

krz nn

This theorem follows from Theorem 4.

For any z ≠ 0, equation (2) has precisely n solutions:

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Geometrical meaning of roots:

To calculate z1/2 where z = –4, draw the following table:

| z | arg z | z1/2 | arg z1/2 = ½ (arg z)

4 π 2 ½ π

4 π + 2π 2 ½ (π + 2π)

).sini(cos2),sini(cos22

3

2

3

2

1

2

1 z

Hence,

i2 i2

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Comment:

You need to memorise the following values of sines and cosines:

θ sin θ cos θ

0 0 1

π /6 1/2 √3/2

π /4 √2/2 √2/2

π /3 √3/2 1/2

π /2 1 0

The symbol √ in the above table denotes square roots.

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Example 9:

Find all roots of the equation w3 = –8 and sketch on the complex plane.

Example 8:

Find: (a) sin 5π/4, (b) cos 2π/3, (c) sin (–5π/6).

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۞ The complex exponential function is defined by

Example 10:

).sini(cosee i yyxyx

Find all z such that Im ez = 0.

Comment:

The polar representation of complex numbers can be re-written in the form

,eirz

where r = | z | and θ = arg z.

(3)

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Comment:

Consider

,ewz

and observe that any value of w corresponds to a single value of z.The opposite, however, isn’t true, as infinitely many values of w (differing from each other by multiples of 2πi) correspond to the same value of z.

This suggests that, even though the exponential is a single-valued function, the logarithm is not.

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۞ The complex number w is said to be the natural logarithm of a complex number z, and is denoted by w = ln z, if

.e zw (4)

Theorem 8:

,)2(Argilnln kzzz

This theorem follows from equalities (3)–(4).

For any z ≠ 0, equation (4) has infinitely many solutions, such that

where k = 0, ±1, ±2, ±3...

(5)

Example 11:

Use (5) to calculate: (a) ln (–1), (b) ln (1 + i).

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۞ The principle value of the logarithm is defined by

.ArgilnLn zzz

۞ General powers of a complex number z are defined by

.e ln zppz

Since this definition involves the logarithm, zp is a multivalued function. It has, however, a single-valued version,

.e of valueprincipal The Ln zppz

Comment:

Evidently, the logarithm, all roots, and non-integer powers of z are all multivalued functions.