By Yue Kuen Kwok

This introductory textual content on complicated variable tools has been up to date with much more examples and exercises.

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4). The proof of the construction is simple. It is seen that α and β lie on the same ray through the origin so that Arg α = Arg β. From the construction, it is observed that |β|/R = R/|α| = sec φ, where φ is the angle between OP and the ray L, so |α| |β| = R 2 . Combining the results, α and β are related by β = R 2 /α. (ii) When |α| = R, the symmetry point is just α itself. (iii) Lastly, when |α| > R, the symmetry point β will be inside the circle CR . To reverse the method of construction in (i), we find a tangent to the circle which passes through α and call the point of tangency P .

A continuous arc without multiple points is called a Jordan arc. If the arc has only one double point, corresponding to the initial and terminal values α and β of t, that is, z(α) = z(β), then it is called a simple closed Jordan arc. Simply connected domains A domain S is said to be simply connected if every simple closed Jordan arc in S can be shrunk to a point in S without passing through points not belonging to S. That is, S is a simply connected domain if for any closed Jordan arc lying in the domain S, the points inside the closed curve also belong to S.

1 In elementary circuit theory, the time-dependent response in an R-L-C series circuit is modeled by the second-order ordinary differential equation L dI 1 d 2I +R + I = 0, dt 2 dt C where I is the instantaneous alternating current flowing through the circuit. Suppose the solution is assumed to be of the form I = Iˆeαt . Find an expression for α in terms of R, L, and C. Show that, as a complex quantity, α is never in the right half complex plane if R, L and C are nonnegative. 36 Complex Numbers Solution Suppose the governing equation admits a solution of the form I = Iˆeαt .