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Unformatted text preview: ECE 493 HW #9 Version 1.31 February 11, 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 &gt; 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 | &gt; | z 2 | ? Solution: | z | = radicalbig x 2 + y 2 is the length of z , so the above expression says that a disk of radius | z 1 | is outside a second disk of radius | z 2 | . (b) How about | z 1 + z 2 | &gt; 3? Solution: In this case we add the two complex numbers together, so this expression says that the total length of the two complex numbers must be greater than 3. If z 1 = 1 and z 2 = i then the sum is 1+ i which has length | 1+ i | = 2, and the condition is violated. (c) If z and w are complex numbers, define the meaning of z &gt; w . Solution: It seems clear that order exists along a line, but does not apply in two dimensions. 2. Analytic functions: State the regions where the following functions are analytic (Note: Im not asking you to apply the CR conditions, just state the region. Remember that the analytic function has a power series that converges in the region of convergence (ROC). Thus an analytic function can be differenciated any number of times. 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 Solution: This is analytic everywhere, so f ( z ) = 2 z everywhere. As noted in class, this already the power series, so taking the derivative may be done term by term. (b) f ( z ) = 1 /z Solution: This is analytic everywhere other than at z = 0. (c) f ( z ) = ln( z ) Solution: Since f ( z ) = 1 / 2 z , the function has a pole at z = 0 thus is analytic except at z = 0. (d) f ( z ) = 1 z 2 Solution: This problem is very similar (let s = iz ) to one in HW8. f ( z ) = 2 z 2 1 z 2 = z f ( z ) . To determine the region where it is analytic, we need to determine where this fails to exists, namely at z = 1. Thus you will need some branch cut(s) between these two points to make the function singular valued. Look in the text on page 1133. (e) Let f ( z ) = n =0 a n z n with a n = 1 (independent of n ). Find f ( z ) and state the region where f ( z ) and f ( z ) are analytic. Solution: Term by term differentiation results in f ( z ) = n =1 nz n . To proceed, one needs to know the ROC of f ( z ). By summing the series, we find that the first pole is at z = 1, thus the ROC is | z | &lt; 1....
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- Spring '11