Download Applied Complex Variables for Scientists and Engineers by Yue Kuen Kwok PDF

By Yue Kuen Kwok

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

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N − 1 . 8) If we write ωn = e2πi/n , then the nth roots are 1, ωn , ωn2 , . , ωnn−1 . Alternatively, if we pick any one of the roots and call it α, then the other roots are given by αωn , αωn2 , . , αωnn−1 . In the complex plane, the nth roots of unity correspond to the n vertices of a regular n-sided polygon inscribed inside the unit circle centered at the origin, with one vertex at the point z = 1. The vertices are equally spaced on the circumference of the circle. 3 shows the regular octagon inscribed inside the unit circle when n = 8.

2 Mapping properties of complex functions A complex function w = f (z) may be considered as the assignment of a point (x, y) in the x-y plane to another point (u, v) in the u-v plane, where z = x + iy and w = u + iv. It is not possible to superimpose these two planes to visualize the graph of f (z). To examine how f (z) works, we put the two planes side by side, select some convenient curves or sets in the x-y plane and then plot their corresponding image curves or image sets in the u-v plane. In this manner, we may consider w = f (z) as a mapping from the x-y plane onto the u-v plane.

In the complex plane, the nth roots of unity correspond to the n vertices of a regular n-sided polygon inscribed inside the unit circle centered at the origin, with one vertex at the point z = 1. The vertices are equally spaced on the circumference of the circle. 3 shows the regular octagon inscribed inside the unit circle when n = 8. The above argument can be extended easily to the nth roots of an arbitrary complex number. Suppose the given complex number in polar form is represented by r(cos φ + i sin φ).

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