1. 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

2. 1. The definition of complex numbers and basic operations
۞ The set of complex numbers can be viewed as the Euclidean vector space R2, of ordered pairs of real numbers (x, y), written as
where i ‘marks’ the second component. x and y are called the real and imaginary parts of z and are denoted by
Like any vectors, complex numbers can be added and multiplied by scalars.
Example 1:
Calculate: (a) (1 + 3i) + (2 – 7i) , (b) (–2)×(2 – 7i).

3. ۞ 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
where r = | z | and θ is the argument of z, or arg z, defined by

4. 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).

5. Example 2:
Show on the plane of complex z the sets of points such that:
(a) | z | = 2, (b) arg z = π/3.

6. 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.
Multivalued functions will be discussed in detail later. In the meantime, we introduce a single-valued version of arg z.
۞ The principal value of the argument of a complex number z is denoted by Arg z (with a capital “A”), and is defined by
The same as arg z

7. Comment:
The graph of w = arg z looks like a spiral staircase in the 3D space (x, y, w). Arg z represents one of its sections.
Arg z

8. Theorem 2: the Triangle Inequality
For any z1 and z2, it holds that | z1 + z2 | ≤ | z1 | + | z2 |.
Proof (by contradiction):
Assume that Theorem 2 doesn’t hold, i.e.
hence...

9. (1)
Our strategy: to get rid of the square roots – cancel as many terms as possible – hope you’ll end up with something clearly incorrect (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
The l.h.s. and r.h.s. of this inequality can be assumed +tive (why?) – hence, we can square it,
This inequality is clearly incorrect (why?) – hence, contradiction. █

10. In addition to the standard vector operations (addition and multiplication by a scalar), complex numbers can be multiplied, divided, and conjugated.
۞ The product of z1 = x1 + i y1 and z2 = x2 + i y2 is given by
Example 4:
Calculate (1 + 3i) (2 – 7i).

11. Remark:
When multiplying a number by itself, one can write z×z = z2, z×z×z = z3, etc.
Example 5:
Observe that or
One cannot, however, write
because the square root is a multivalued function (more details to follow).

12. ۞ Complex numbers z1 = x + iy and z2 = x − iy are called complex conjugated (to each other) and are denoted by or
Example 6:
If z = 5 + 2i, then z* = 5 – 2i.

13. Theorem 3:
Proof: by direct calculation.

14. ۞ The quotient z =z1/z2, where z2 ≠ 0, is a complex number such that
Example 7:
Calculate (1 + 3i)/(2 – 7i).

15. cos θ1 cos θ2 – sin θ1 sin θ2 = cos (θ1 + θ2).
Theorem 4: multiplication of complex numbers in polar form
where r1,2 = | z1,2 | and θ1,2 = arg z1,2.
Proof: by direct calculation.
Useful formulae:
sin θ1 cos θ2 + sin θ2 cos θ1 = sin (θ1 + θ2),
cos θ1 cos θ2 – sin θ1 sin θ2 = cos (θ1 + θ2).

16. Theorem 5:
where r = | z | and θ = arg z.
This theorem follows from Theorem 4 with r1 = r2, which you need to apply n times.
Comment:
In what follows, we’ll often use r and θ in the meaning of | z | and arg z, respectively.
Theorem 6: The de Moivre formula
This theorem follows from Theorem 5 with r = 1.

17. 2. Roots, the exponential function, and the logarithm
۞ The nth root of a complex number z is a complex number w such that
(2)
The solutions of equations (2) are denoted by w = z1/n.
Theorem 7:
For any z ≠ 0, equation (2) has precisely n solutions:
where r = | z |, θ = Arg z, and k = 0, 1... n – 1.
This theorem follows from Theorem 5.

18. r = | z | | w | = r1/2 4 2 Arg z k arg w = ½ (Arg z + 2πk) π ½ π 4 2 π
Geometrical meaning of roots:
To calculate w = z1/2 where z = –4, draw the following table:
r = | z |
| w | = r1/2 4 2
Arg z k
arg w = ½ (Arg z + 2πk) π
½ π 4 2 π 1
½ (π + 2π)
Hence,

19. θ sin θ cos θ 1 π /6 1/2 √3/2 π /4 √2/2 π /3 π /2 Comment:
It’ll be helpful to remember the following values of sines/cosines: θ
sin θ
cos θ 1
π /6
1/2
√3/2
π /4
√2/2
π /3
π /2
The symbol √ in the above table denotes square roots.

20. Example 8:
Find: (a) sin 5π/4, (b) cos 2π/3, (c) sin (–5π/6).
Example 9:
Find all roots of the equation w3 = –8 and sketch on the complex plane.

21. ۞ The complex exponential function is defined by
Example 10:
Find all z such that Im ez = 0.
Comment (a very important one!):
The polar representation of complex numbers can be re-written in the form
(3)
where r = | z | and θ = arg z.

22. Comment:
Consider
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, e.g.
This suggests that, even though the exponential is a single-valued function, the logarithm is not.

23. ۞ The complex number w is said to be the natural logarithm of a complex number z, and is denoted by w = ln z, if
(4)
Theorem 8:
For any z ≠ 0, equation (4) has infinitely many solutions, such that
(5)
Since arg z is a multivalued functions, so is ln z.
Example 11:
Use (5) to calculate: (a) ln (–1), (b) ln (1 + i).

24. ۞ The principle value of the logarithm is defined by
۞ General powers of a complex number z are defined by
Since this definition involves a logarithm, zp is a multivalued function. It has, however, a single-valued version,