Show that any complex number s satisfying | s | 1 yields
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Show that any complex number s satisfying | s | 1 yields a value of z s s 1 1 that satisfies Re(z) 0 ( Hint: Let s x jy;
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Show that any complex number s satisfying | s | 1 yields a value of z s s 1 1 that satisfies Re(z) 0 ( Hint: Let s x jy; z u jv. Rationalize the fraction, and equate real and imaginary parts of z and the rationalized fraction. Now consider what happens to the circle x2 y2 1. To show that the inside of the circle goes over to the right half-plane, consider a convenient point inside the circle.) On the basis of this transformation, deduce an extension of the Routh criterion that will determine whether the system has roots inside the unit circle. Why might this information be of interest? How can the transformation be modified to consider circles of other radii?
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