Complex Numbers. w = f(z) z. Examples

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1 omple Numbers Geometrical Transformations in the omple Plane For functions of a real variable such as f( sin, g( 2 +2 etc ou are used to illustrating these geometricall, usuall on a cartesian graph. If we have functions of a comple variable given b equations such as w sin z or w z 2 +2 we cannot use a cartesian graph, since z cannot be represented on an ordered ais. Indeed z ma range over the whole of the two dimensional comple plane, so that if w is also comple we would need a 4-dimensional space to plot a graph such as w z Most of us cannot visualise this, and what we usuall do is to have two copies of the comple plane, and we look at points in the z-plane and see how the are transformed into points in the w-plane. We also look at sets of points, curves or regions in the z-plane and their images in the w-plane. w w f(z z Eamples 1 w f(z z + 2. This simpl shifts ever point two units in the direction of the real ais - it is a translation. z z + 2 1

2 2 w z + 2 i, again a translation z z + 2 i 3 w z + 2, this is not a translation. z z w 2z Now w 2 z arg w arg 2 + arg z arg z So this is an enlargement about the origin with scale factor 2. z 2z 2

3 5 w iz w z arg w arg i + arg z π 2 + arg z So this is a rotation through π 2 anticlockwise about O. z iz In general if α is an comple number and we write α re iθ then w αz is an enlargement b scale factor r together with a rotation about O through the angle θ anticlockwise. If we write then w αz becomes u + iv (a + ib( + i and so α a + ib z + i w u + iv u a b v b + a u a b We write this in the form v b a The right hand side can be interpreted as a multiplication, but at the moment it seems( a rather odd kind of multiplication. We call a column vector. a b We call a matri. b a If we now have another transformation ξ βw where β c + id then if we write ξ s + it we shall have ( s t ( c d d c ( c d d c u v ( a b b a ( 3

4 If we now do the substitutions s cu dv t du + cv in the first pair of equations we get s (ca db (cb + da t ((ad + bc + (ac bd s (ca bd (cb + da t (ad + bc (ac bd This ( suggests ( that we should ( define c d a b (ca bd (cb + da d c b a (ad + bc (ac bd Finall if we go back to the original equation w αz v βw we obtain ξ βαz and βα (c + id(a + ib (ac bd + i(ad + bc If we write α and β in polar form, taking r 1 for both, so that the both correspond to rotations, we then have α cos θ + i sin θ β cos φ + i sin φ The ( corresponding ( matrices are cos θ sin θ cos φ sin φ sin θ cos θ sin φ cos φ cos θ cos φ sin θ sin φ (cos θ sin φ + sin θ cos φ sin θ cos φ + cos θ sin φ cos θ cos φ sin θ sin φ cos(θ + φ sin(θ + φ sin(θ + φ cos(θ + φ which is in accordance with what we found previousl. Notice that although ( each comple number can be represented b a matri, 1 1 matrices such as do not correspond to comple numbers. We can 0 1 ( nevertheless ( use them ( to transform the plane This corresponds to a shearing transformation. 4

5 In considering matrices used as transformations we have so far considered the ( problem ( of finding the image of given points. X Y X i.e. given what is? Y We now ( consider the reverse problem: X given what is? Y a b X c d Y so a + b X (1 c + d Y (2 (1 d and (2 b ad + bd dx bc + bd by subtracting gives (ad bc dx by (3 (1 c and (2 a ac + bc cx ac + ad ay subtracting gives (ad bc ay cx (4 (3 and (4 can be solved for and iff ad bc 0. If ad bc 0 we then have d ad bc X b ad bc Y c ad bc X + a ad bc Y so 5

6 ( ( d b ad bc ad bc X c a Y ad bc ( ad bc 1 d b X ad bc c a Y d b 1 The matri ( d c b a c a a b is called the inverse of written c d s a transformation this matri does nothing at all. ll points are fied. It is called the identit matri. ad bc is called the determinant of. So has an inverse iff its determinant is non-zero. For a( comple number matri a b a α 2 + b 2 α 2 b a 0 iff a b 0 i.e. α 0 and its ( inverse is 1 a b α α 2 b a α 1 α 0 2 α In widening the sstem to include all possible 2 2 matrices we have included man matrices which do not have inverses. We have also sacrificed commutativit of multiplication, as does not alwas equal. However we can deal with man different transformations, and matrices turn out to have man and varied applications. Other transformations There are man transformations not represented b 2 2 matrices as above. s an eample we consider a few properties of the transformation w z 2. It is convenient to use polar co-ordinates, we use (r, θ in the z-plane and (p, φ in the w-plane. 6

7 z-plane w-plane E z re iθ z 2 r 2 e 2iθ so p r 2 φ 2θ E (1, 0 (1, 0 ( 2, π 4 (2, π 2 (1, pi (1, π 2 (1, π (1, 2π (1, 0 E(1, 3π E (1, 3π (1, π 2 IGRM so z e iθ π θ π corresponds to a circle traced twice in the w-plane. IGRM z-plane w-plane so z re iθ 0 < θ < π w r 2 e 2iθ 0 < 2θ < 2π pe iθ 0 < φ < 2π upper half plane > 0 plane without +ve real ais Reverting to cartesians now let z + i w ξ + iη ξ + iη i so ξ 2 2 η 2 Now if 1, ξ 1 2 η 2 so ξ 1 η2 4 IGRM If 1 ξ 2 1 η 2 so ξ η2 4 1 IGRMS 7

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