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HW9 - ECE 493 Univ of Illinois HW#9 Version 1.21 Due Tu Feb...

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ECE 493 HW #9 – Version 1.21 January 27, 2011 Spring 2011 Univ. of Illinois Due Tu, Feb. 1 Prof. Allen Topic of this homework: Analytic functions: Integration of analytic functions; Cauchy integral formula; Riemann Sheets and Branch cuts; Region of Convergence; inverse Laplace transforms; Deliverable: Show your work. In this homework i = - 1. 1. Ordering complex numbers One can always say that 3 > 4, namely that real numbers have order . We will explore if complex numbers have order. Let z = x + iy be a complex number. (a) Can you define a meaning to | z 1 | > | z 2 | ? (b) How about | z 1 + z 2 | > 3? (c) If z and w are complex numbers, define the meaning of z > w . 2. Analytic functions: State the regions where the following functions are analytic (Note: I’m not asking you to apply the CR conditions, just state the region. Try to expand the function is a power series, and then look for the ROC. Consider also the expansion of df ( z ) /dz . (a) f ( z ) = z 2 (b) f ( z ) = 1 /z (c) f ( z ) = log( z ) (d) f ( z ) = 1 - z 2 (e) Let f ( z ) = n =0 a n z n . Find f ( z ) and state the region where f ( z
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