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MA3111 Complex Analysis I: Lecture Notes NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics
Prelude Historically, people were reluctant to use new number systems unless they found them necessary. After all, the world is real, so it seems that real numbers are enough. Why bother to introduce imaginary numbers? Ancient Greeks did not mention complex numbers. (Pythagoras even did not like irrational numbers 1 .) Medieval mathematicians thought that they could never surpass ancient Greeks, so they naturally would not introduce anything they did not learn from old codices. The renaissance of mathematics began with the solution to cubic equa- tions, which ancient Greeks had failed. Let the cubic equation be ax 3 + bx 2 + cx + d = 0 , a 6 = 0 . (Of course we assume all coefficients real.) Define Δ 0 = b 2 - 3 ac, Δ 1 = 2 b 3 - 9 abc + 27 a 2 d, C = 3 s Δ 1 ± p Δ 2 1 - 3 0 2 , Here the ± sign can be arbitrary. Then the general formula of the three roots are x k = 1 3 a b + ξ k C + Δ 0 ξ k C , k = 0 , 1 , 2 , where ξ = - 1 2 + 1 2 3 i is a complex number, and actually is a cube root of unity: ξ 3 = 1. From our experience, we know that many cubic equations have three real roots. But the solution formula is given in the meaningful form only if we allow complex numbers — although the final result may turn out to be all real. This fact made complex numbers an indispensable part of modern math- ematics. 1 A myth is discussed in i
CHAPTER 1 Complex Numbers 1.1. Complex Numbers Let R denote the set of real numbers. However, there are numbers lying outside R . For example, i = - 1 / R . One can reason this as follows: For any number α R , one has α 2 0; but i 2 = - 1 < 0. Definition 1.1.1 . Complex numbers are numbers of the form z = x + iy with x, y R . Example 1.1.2 . 2 i , πi , 2 πi and 1 + i are complex numbers. We denote the set of complex numbers by C , that is C = { x + iy | x, y R } . Remark 1.1.3 . For z = x + iy , (1) x is called the real part of z , and y is called the imaginary part of z . We usually write x = Re z, y = Im z. (2) If y = 0, then z = x + i 0 is real . If x = 0, then z = 0 + iy is said to be purely imaginary . (3) We will identify the set of real numbers as a subset of the set of complex numbers (i.e. R C ) under the identification x ←→ x + i 0 for x R . (4) Equality of two complex numbers: Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. In other words, if z 1 = x 1 + iy 1 and z 2 = x 2 + iy 2 , then z 1 = z 2 ⇐⇒ x 1 = x 2 and y 1 = y 2 . Example 1.1.4 . (1) Re(2 + 3 i ) = 2 , Im(2 + 3 i ) = 3 , Im( - 2 i + 5) = - 2 . (2) The imaginary parts of - 5, 1+2 i and ei are 0, 2 and e respectively. (3) Solve the equation 4 + xi = ( y 2 - 5) + 3 i, where x, y R . 1
2 1. COMPLEX NUMBERS Solution. 4 + xi = ( y 2 - 5) + 3 i ⇐⇒ 4 = y 2 - 5 and x = 3 ⇐⇒ y 2 = 9 and x = 3 ⇐⇒ y = ± 3 and x = 3 ⇐⇒ ( x, y ) = (3 , 3) or (3 , - 3) . 1.2. Algebraic Operations and Properties Definition 1.2.1 (addition/subtraction, multiplication) . For z 1 = x 1 + iy 1 and z 2 = x 2 + iy 2 , one defines ( x 1 + iy 1 ) + ( x 2 + iy 2 ) := ( x 1 + x 2 ) + i ( y 1 + y 2 ) , ( x 1 + iy 1 ) - ( x 2 + iy 2 ) := ( x 1 - x 2 ) + i ( y 1 - y 2 ) , ( x 1 + iy 1 ) · ( x 2 + iy 2 ) := ( x 1 x 2 - y 1 y 2 ) + i ( x 1 y 2 + y 1 x 2 ) .
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