Yue Kuen Kwok's Applied Complex Variables for Scientists and Engineers PDF

Calculus

By Yue Kuen Kwok

ISBN-10: 0511775008

ISBN-13: 9780511775000

ISBN-10: 0521701384

ISBN-13: 9780521701389

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

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Example text

The two examples presented below illustrate some interesting mapping properties of complex functions. 3 The complex numbers z = x + iy and w = u + iv are represented by points P and Q in the z-plane and w-plane, respectively. Suppose P traverses along the vertical line x = − 12 , and z and w are related by z= w . 1−w Find the trajectory traced out by Q. Solution It is convenient to express the mapping relation as z w= . 1+z Since P moves along x = − 12 , we may represent P by z = − 12 + iy, where y can take any real value.

4 that Suppose the four points z1 , z2 , z3 and z4 lie on a circle. Show (z1 − z3 )(z2 − z4 ) (z1 − z4 )(z2 − z3 ) is real. 6. Angles of turning of the line segments at z1 and z2 are equal, that is, θ1 = θ2 . 6. Draw the two line segments, one joining z1 with z3 and the other joining z1 with z4 . The angle of turning of 4 . 6) is given by Arg zz11 −z −z3 Similarly, we draw another pair of line segments joining z2 with z3 and z2 with z4 . 6) 4 . Since the four points are concyclic, the two angles of is given by Arg zz22 −z −z3 turning are equal.

1 Derive a formula for the length of the chord joining the images on the Riemann sphere corresponding to the two points z1 and z2 in the complex plane. Examine the case when one of the points is the point at infinity. Solution Let the corresponding image points of z1 and z2 on the Riemann sphere be P1 (ξ1 , η1 , ζ1 ) and P2 (ξ2 , η2 , ζ2 ), respectively. Using eqs. 3a,b,c), we have d(z1 , z2 )2 = (ξ1 − ξ2 )2 + (η1 − η2 )2 + (ζ1 − ζ2 )2 = (ξ12 + η12 + ζ12 ) + (ξ22 + η22 + ζ22 ) − 2(ξ1 ξ2 + η1 η2 + ζ1 ζ2 ) = (ζ1 + ζ2 ) − 2(ξ1 ξ2 + η1 η2 + ζ1 ζ2 ) |z1 |2 |z2 |2 x1 x2 + y1 y2 + |z1 |2 |z2 |2 = + − 2 1 + |z1 |2 1 + |z2 |2 (1 + |z1 |2 )(1 + |z2 |2 ) 2 |z1 − z2 | .

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Applied Complex Variables for Scientists and Engineers by Yue Kuen Kwok


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