Complex Numbers Why shall we study complex analysis? We list several examples to illustrate why shall we study complex analysis. (Algebra) If we only limit ourselves to real numbers, the quadratic equation where a 0,b,c R has root ax +bx+c = 0, x = b± b 4ac a provided b 4ac 0. However, if we allow complex numbers, the equation always has exactly two (complex) roots. More generally, the fundamental theorem of algebra proved by Gauss asserts that for any polynomial equation a n z n + a n z n +... + a z + a 0 = 0, where a j C with a n 0, always has exactly n complex roots. (Analysis) The concept of infinity was a mystery for thousand years in the history of mathematics. Only after Cauchy s work in the 9th century, people began to understand how to deal with the infinity in analysis. Even in modern calculus books, the definitions for lim a n = A, A, and lima n = seem to be different. However, if we use the notion of n n Riemann Sphere or the complex projective space, then the infinity can be treated as an ordinary point (i.e., a complex number), and there the definitions of lima n = A, n A, and lima n = are the same. n (Physics General relativity and differential geometry) The most popular version of String theory requires the universe to be 0-dimensional for this quantization process to work. The space we live in looks locally like 3 M = R 4 X where R 4 is the Minkowski space and X is a compact Calabi-Yau 3 dimensional complex manifold with radius of order 0 33 cm, the Planck length. Since the Planck length is so small, space then appears to macroscopic observers to be 4-dimensional. The reason why X has a complex structure because of the supersymmetry in this model. See Lecture 3 for more details. See Lecture 5 for more details. 3 M. Gross, D. Huybrechts and D. Joyce, Calabi-Yau Manifolds and Related Geometries, p.63. 4
(The number theory) Let {p j } j= be a sequence of increasing all prime numbers (i.e., p =,p = 3,p 3 = 5,...). By a formula for the geometric series 4, if z > 0 is a positive number, we have p z = + p z + (p ) z + (p 3 ) z +... = n z j=,j, where {n,j } is the set of all integers of the form p i for all integers i 0. Also, ( p z p z = + + + )( +... + + + ) +... p z p z p 3z p z p z p 3z = + (p p ) + z (p p ) + z (p p + )z (p +... = p )z n z j=,j where {n,j } is the set of all integers of the form p i p i for all integers i,i 0.... By repeating this process, one gets q k= ( ) p z = k n z j= q,j where {n q,j } istheset ofall integersof theformp i p i... pq iq forallintegers i,i,...,i q 0. Let q, we obtain n= ( ) = p z n This is Euler s formula which gives a connection between the number theory and the complex analysis. The right-hand-side is the well-known function, denoted as ζ(z) and called Riemann zeta function, in which the variable z could be a complex number. The famous unsolved Riemann Hypothesis asserts that all of the zero points p of the function ζ(z) must satisfy Re(p) =. The above Euler formula gives an immediate proof of the fact that the set of all prime numbers cannot be finite. The origin of complex numbers Where did the notion of complex numbers came from? Did it come from the equation x + = 0 () as i = is defined today? 4 r = +r+r +r 3 +... for r <. 5 n= n z.,
No. Very long times ago people had no problem to accept the fact that an equation may have no solution. When Brahmagupta (598-668) introduced a general solution formula x = b± b 4ac a for the quadratic equation ax +bx+c = 0, he only recognized positive real root. The starting point of emerging the notion of complex number indeed came from the theory of cubic equations. In the 6th century, cubic equations was solved by the del Ferro- Tartaglia-Cardano formula: a general cubic equation can be reduced into a special one: y 3 = py +q and this equation can be solved by y = 3 q + ( q ) ( p) 3 q + 3 3 ( q ) ( p) 3. () 3 Cardano (50-576) was the first to introduce complex numbers a+ b into algebra, but he had misgivings about it. He made a comment that dealing with involves mental tortures and is truly sophisticated and these numbers were as subtle as they are useless. 5 Here comes a problem. A cubic equation y 3 py q = 0 with reall coefficients p and q always has a real solution y because we can consider the graph: y 3 py q > 0 as y is a large positive number and y 3 py q < 0 as y is a large negatively number so that the graph curve must intersect the x-axis. On the other hand, the the number inside the square root in (), ( ) q ( p 3 3), could be negative. How could the formula () produce a real solution in this case? Concerning on this problem, in 569, Rafael Bombelli (56-57) observed that the cubic equation x 3 = 5x+4 does have a root x = 4, but by the formula () gives x = 3 + + 3. (3) Bombelli tried to give an explanation. He put 3 + = a+ib from which he deduced 3 = a ib and obtained a = and b = so that x = a+bi+a bi = a = 4. He might consider the equations + = (a + ib) 3 and = (a ib) 3 and found that (a,b) = (,) is a solution. 5 J.H. Mathews and R.W. Howell, Complex Analysis for Mathematics and Engineering, 5th ed., Jones and Bartlett Publishers, 006, p.3. 6
This is the first time that the notion of complex number appeared. However, Bombelli did not really understand it. After doing this, Bombelli commented: At first, the thing seemed to me to be based more on sophism than on truth, but I searched until I found the proof. Rene Descartes (596-650), who was a pioneer to work on analytic geometry and used equation to study geometry, called complex numbers impossible. In fact, the terminology imaginary number came from Descartes. Issac Newton (643-77) agreed with Descartes. He wrote: it is just that the roots of equations should be often impossible (complex), lest they should exhibit the cases of problems that are impossible as if they were possible. 6 Gottfried WiIhelm Leibniz (646-76), who and Newton established calculus, remarked that theimaginaryfactorsofx 4 +a 4 as anelegant andwonderful recourse ofdiving intellect, an unnatural birth in the realm of thought, almost an amphibium between being and nonbeing. 7 As time passes, mathematicians gradually redefined their thinking and began to believe that complex numbers existed, and set out to make them understood and accepted. Wallis tried in 673 to give a geometric representation which failed but was quite close. Johann Bernoulli noted in 70 that from which he used integral to get dz +z = dz (+z ) + dz ( z ) tan z = i logi z i+z. Around 740, Leonhard Euler(707-783) defined the exponential e x := +x+! x + 3! x3 +... and proved the famous identity e iθ = cos θ+i sin θ. In his proof, Euler also used the formulas sin x = x 3! x3 + 5! x5... and cos x =! x + 4! x4... In 988, readers of the Mathematical Intelligence voted it the Most 6 Morris Kline, Mathematical Thought from Ancient to Modern Times, volume, New York Oxford, Oxford University Press, 97, p.54. 7 F. Cajori, Historyofthe exponentialand logarithmicconcepts, AmericanMath Monthly, vol. XX(93), p.39. 7
Beautiful Mathematical Formula Ever. In total, Euler was responsible for three of the top five formulas in that poll. 8 In 799 Carl Friedrich Gauss gave the first of his four proofs for the well-known Fundamental Theorem of Algebra: Any polynomial equation has exactly n complex roots. a n x n +a n x n +...+a x+a 0 = 0 (a n 0) In 8 Gauss wrote to Bessel to indicate that many properties of the classical functions are only fully understood when complex arguments are allowed. In this letter, Gauss described the Cauchy integral theorem but this result was unpublished. In 799 Caspar Wessel (745-88) first gave the geometrical interpretation of complex numbers z = x+iy = r(cos θ +isin θ) where r = z and θ R is the polar angle. Wessel s approach used what we today call vectors. He uses the geometric addition of vectors (parallelogram law) and defined multiplication of vectors in terms of what we call today adding the polar angles and multiplying the magnitudes. Wessel s paper, written in Danish in 797, went unnoticed in most of Europe until 897, when it was unearthed by an antiquarian, and its significance recognized by the Danish mathematician Sophus Christian Juel. The same fate awaited the similar geometric interpretation of complex numbers put forth by the Swiss bookkeeper J. Argand (768-8) in a small book published in 806. It was only because Gauss used the same geometric interpretation of complex numbers in his proofs of the fundamental theorem of algebra and in his study of quartic residues that this interpretation gained acceptance in the mathematical community. 9 William Rowan Hamilton (805-65) in an 83 memoir defined ordered pairs of real numbers (a,b) to be a couple. He defined addition and multiplication of couples: (a,b) + (c,d) = (a+c,b+d) and (a,b)(c,d) = (ac bd,bc+ad). This is the first algebraic definition of complex numbers. Definition of Complex Numbers Since the equation x + = 0 8 Here is the list of the top five:. Euler s formula;. Euler s formula for a polyhedron: V +F = E +. 3. The number of primes is infinite. 4. There are 5 regular polyhedrons. 5. + + 3 + 4 +... = π 6 (Euler). cf., David Wells, Are these the most beautiful? Mathematical Intelligencer (3)(990), 37-4. 9 Victor J. Katz, A History of Mathematics - an introduction, 3rd edition, Addison -Wesley, 009, p.796. 8
admits no real solution, we define i to be a formal solution: i + = 0. We define a complex number to be z = x+iy where x,y R are real numbers. Re(z) := x is called the real part of z, and Im(z) := y is called the imaginary part of z. Then we define the space of complex numbers C := {z = x+iy x,y R}. For two complex numbers z = x+iy and w = u+iv, we define. (equality) z = w x = u and y = v.. (addition) z +w = (x+u)+i(y +v). 3. (multiplication) zw = (x+iy)(u+iv) = (xu yv)+i(xv +yu). 0 4. (division) w = z w, where z 0 and z z := x+iy = x iy (x+iy)(x iy) = x iy x +y = As a result, the set of all complex numbers C is a field. [Example] Find the value of 5 3+4i. Solution: 5 3+4i = 5( 3 4i) ( 3+4i)( 3 4i) = 5 0i 9+6 x x +y i y x +y. = 5+0i. 5 0 Here we used the fact that i =. 9