Mathematical Methods for Physics and Engineering - Matematica.NET

COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONSIm zyz = x + iyxRe z−yz ∗ = x − iyFigure 3.6The complex conjugate as a mirror image in the real axis.Inthecasewherez can be written in the form x + iy it is easily verified, bydirect multiplication of the components, that the product zz ∗ gives a real result:zz ∗ =(x + iy)(x − iy) =x 2 − ixy + ixy − i 2 y 2 = x 2 + y 2 = |z| 2 .Complex conjugation corresponds to a reflection of z in the real axis of theArgand diagram, as may be seen in figure 3.6.◮Find the complex conjugate of z = a +2i +3ib.The complex number is written in the standard formz = a + i(2+3b);then, replacing i by −i, weobtainz ∗ = a − i(2+3b). ◭In some cases, however, it may not be simple to rearrange the expression forz into the standard form x + iy. Nevertheless, given two complex numbers, z 1and z 2 , it is straightforward to show that the complex conjugate of their sum(or difference) is equal to the sum (or difference) of their complex conjugates, i.e.(z 1 ± z 2 ) ∗ = z1 ∗ ± z∗ 2 . Similarly, it may be shown that the complex conjugate of theproduct (or quotient) of z 1 and z 2 is equal to the product (or quotient) of theircomplex conjugates, i.e. (z 1 z 2 ) ∗ = z1 ∗z∗ 2 and (z 1/z 2 ) ∗ = z1 ∗/z∗ 2 .Using these results, it can be deduced that, no matter how complicated theexpression, its complex conjugate may always be found by replacing every i by−i. To apply this rule, however, we must always ensure that all complex parts arefirst written out in full, so that no i’s are hidden.90