Week 1

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 Week 1

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 yare called the real and imaginary parts of z and are denoted 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 + iy, 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: A complex number z = x + iy 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 the number 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 Argz (with a capital “A”), and is defined by

7. Comment: The graph of argzlooks like a spiral staircase. Argz

8. Theorem 2: the Triangle Inequality For any z1and 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: 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, 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 + iy1 and z2 = x2 + iy2 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. ۞ The quotient z =z1/z2, where z2 ≠ 0, is a complex number such that Example 6: Calculate (1 + 3i)/(2 – 7i).

13. Theorem 3: 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).

14. Theorem 4: where r = | z | and θ = Arg z. This theorem follows from Theorem 3. Theorem 5: The de Moivre formula This theorem follows from Theorem 4 with | z | = 1.

15. ۞ Complex numbers z1 = x + iy and z2 = x − iy are called complex conjugated (to each other) and are denoted by or Example 7: If z = 5 + 2i, then z* = 5 – 2i.

16. Theorem 6: Proof: by direct calculation.

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

18. Theorem 7: For any z ≠ 0, equation (2) has precisely nsolutions: where r = | z |,θ = Arg z, andk = 0, 1... n – 1. This theorem follows from Theorem 4.

19. Geometrical meaning of roots: To calculate z1/2 wherez = –4, draw the following table: Hence,

20. Comment: You need to memorise the following values of sines and cosines: The symbol √ in the above table denotes square roots.

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

22. ۞ The complex exponential function is defined by (3) Example 10: Find all z such that Imez = 0. Comment: The polar representation of complex numbers can be re-written in the form where r = | z | andθ = arg z.

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

24. ۞ The complex number w is said to be the natural logarithm of a complex number z, and is denoted by w= lnz, if (4) Theorem 8: For any z ≠ 0, equation (4) has infinitely many solutions, such that (5) where k = 0, ±1, ±2, ±3... This theorem follows from equalities (3)–(4). Example 11: Use (5) to calculate: (a) ln(–1), (b) ln(1 + i).

25. ۞ The principle value of the logarithm is defined by ۞ General powers of a complex number z are defined by Since this definition involves the logarithm, zp is a multivalued function. It has, however, a single-valued version, Comment: Evidently, the logarithm, all roots, and non-integer powers of z are all multivalued functions.