Chapter 1 From i to z: the basics of complex analysis 1.1. The field of complex numbers The field C of complex numbers is obtained by adjoining i to the field R of reals. The defining property of i is i2 + 1 = 0 and complex numbers z1 = x1 + iy1 and z2 = x2 + iy2 are added componentwise and multiplied according to the rule z1 · z2 = x1x2 y1y2 + i(x1y2 + x2y1), which follows from i2 + 1 = 0 and the distributional law. The complex conjugate of z = x + iy is ¯ = x iy and we have |z|2 := = x2 + y2. Therefore, every z = 0 has a multiplicative inverse given by 1 z := ¯|z|−2 and C becomes a field. Since complex numbers z can be represented as points or vectors in R2 in the Cartesian way, we can also assign polar coordinates (r, θ) to them. By definition, r = |z| and z = r(cos θ + i sin θ). The addition theorems for cosine and sine imply that z1 · z2 = |z1||z2|(cos(θ1 + θ2) + i sin(θ1 + θ2)), which reveals the remarkable fact that complex numbers are multiplied by multiplying their lengths and adding their angles. In particular, |z1z2| = |z1||z2|. This shows that power series behave as in the real case with respect to convergence, i.e., n=0 anzn converges on |z| R and diverges for every |z| R, where R−1 = lim sup n→∞ |an| 1 n , 1
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