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Get Applied Complex Variables for Scientists and Engineers PDF

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 .