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# I s is called the domain of f examples 22 5 a define f

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Unformatted text preview: = ı x |§(x,t)|2 dx . 0 Calculate “x‘ and compute its amplitude of oscillation. 2. Functions of a Complex Variable: Analytic Functions and the Cauchy-Riemann Equations) (§§17.4, 17.5 in Zill) Definition 2.1 Let S ¯ C A complex valued function on S is a function I f: S ’ C . I S is called the domain of f. Examples 2.2 5 (a) Define f: C ÆC by f(z) = z2; II 1 + z. Find g(1+i). – z (c) Define h: C ÆC by h(x+iy) = x + i(xy). II (b) Define g: C -{0}ÆC by g(z) = I I Notes (a) In general, a complex valued function is completely specified by its real and imaginary parts. For example, in (a) above, f(x+iy) = (x+iy)2 = (x2-y2) + i(2xy). Write this as u(x,y) + iv(x,y), where u(x,y) and v(x,y) are a pair of real-valued functions. (b) An important way to picture a function f: S ’C is as a “mapping” - picture in I class. Examples 2.3 (a) Look at the action of the functions z + z0 and åz for fixed z0 é C and å real. I 1 (b) Let S be the unit circle; S = S = {z : |z| = 1}. Then the functions f: SÆS; f(z) = zn ar...
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## This document was uploaded on 03/20/2014 for the course MATH 144 at Hofstra University.

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