Transcription of 1 Basics of Series and Complex Numbers
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1 Basics of Series and Complex Numbers
C FW Math 321, 2012/12/11 Elements of Complex Calculus 1 Basics of Series and Complex Numbers Algebra of Complex Numbers A Complex number z = x + iy is composed of a real part <(z) = x and an imaginary part =(z) = y, both of which are real Numbers , x, y R. Complex Numbers can be defined as pairs of real Numbers (x, y) with special manipulation rules. That's how Complex Numbers are defined in Fortran or C. We can map Complex Numbers to the plane R2 with the real part as the x axis and the imaginary part as the y-axis. We refer to that mapping as the Complex plane. This is a very useful visualization. The form x + iy is convenient with the special symbol i standing as the imaginary unit defined such that i2 = 1. With that form and that special i2 = 1 rule, Complex Numbers can be manipulated like regular real Numbers . y z = x + iy |z|. x z = x i y Addition/subtraction: z1 + z2 = (x1 + iy1 ) + (x2 + iy2 ) = (x1 + x2 ) + i(y1 + y2 ). (1). This is identical to vector addition for the 2D vectors (x1 , y1 ) and (x2 , y2 ).
The geometric series leads to a useful test for convergence of the general series X1 n=0 a n= a 0 + a 1 + a 2 + (12) We can make sense of this series again as the limit of the partial sums S n = a 0 + a 1 + + a n as n!1. Any one of these nite partial sums exists but the in nite sum does not necessarily converge. Example: take a
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