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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**Extra info for Applied Complex Variables for Scientists and Engineers**

**Example text**

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).