By Yue Kuen Kwok

ISBN-10: 0511775008

ISBN-13: 9780511775000

ISBN-10: 0521701384

ISBN-13: 9780521701389

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

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

**Example text**

E. fluid flowing in) at z = β in the complex plane, both of the same strength k. Find the resulting velocity at an arbitrary point z. What happens if the source and the sink are approaching each other such that (α − β) → 0 and k → ∞ but µ = k(α − β) is kept finite? Solution From eq. 4), the velocities at z due to the source and the sink are given by k z−α and − k z−β , respectively. Assuming that the superposition principle of velocities is applicable, the combined velocity at z is given by the sum of the two velocity functions, so v(z) = k 1 1 − z−α z−β = k(α − β) (z − α) (z − β) .

I) To prove the other half of the triangle inequality, we write |z1 | = |(z1 + z2 ) + (−z2 )| ≤ |z1 + z2 | + | − z2 |, giving |z1 | − |z2 | ≤ |z1 + z2 | . (ii) By interchanging z1 and z2 in the above inequality, we have |z2 | − |z1 | ≤ |z1 + z2 | . (iii) Combining the results in eqs. (i), (ii) and (iii), we obtain |z1 | − |z2 | ≤ |z1 + z2 | ≤ |z1 | + |z2 | . 2 Find the locus of the points z satisfying |z − α| + |z + α| = 2r , where α is complex and r is real. Discuss various possibilities and find the maximum and minimum values of |z|.

2 Show that any small circle whose circumference contains the north pole N on the Riemann sphere corresponds to a straight line not passing through the origin in the complex plane. Solution A small circle whose circumference contains the north pole N on the Riemann sphere is determined by the intersection of the Riemann sphere ξ 2 + η2 + ζ 2 − ζ = 0 and a plane through N . In terms of ξ, η and ζ , a plane passing through the north pole N (0, 0, 1) has the general form Aξ + Bη + C(ζ − 1) = 0, C = 0.

### Applied Complex Variables for Scientists and Engineers by Yue Kuen Kwok

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