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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. Week 1. 1. The definition of complex numbers and basic operations.

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Week 1

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  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.

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