Page 14 - Calculus with Complex Numbers

Ffg ure I.2

Ffg ure I.3

I .6 Polar form

An alternative representation of points inthe plane is by polar coordinates r 0. The
coordinate r represents the distance of the point from the origin 0. The coordinate
0 represents the angle the linejoining the point to O makes with the positive direc-
tion of the ar-axis measured anticlockwise (see Figtlre 1.3). Suppose the complex
number z = .x + iy on the Argand diagram has polar coordinates r 0. W e call r

the modulus of z, and denote it by Iz I. Pythagoras' theorem gives

121 = .12 + y2

consistent with the delinition of Izlgiven in Section 1.2.
W e call 0 the argument of z which we abbreviate to arg z. A little trigonometl'y
on Figure 1.3 gives
0 - 1 JF -1 JF -1 .Y
= tan - = sin - = cos -.
.z7 r r
Observe that whilst I zl is single valued arg z is many valued. This is because
for any given value of 0 we could take instead 0 + 2:7r (in radians) and arrive at
the same complex number z. For example, suppose z = 1 + i . Then I z I= .vC,

but arg z can be taken to be any of the values zr/4 5>/4 9>/4 etc. also -3>/4
7>/4 etc. Hquivalently, arg z = zr/4 + lnn' for any integer rl.
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We deline tjmprincipal value (PV) of arg z to be that value of 0 which satislies
zr < 0 :jq zr . For example, the principal value of arg (1 + ï) is zr/4 (Figure 1.4).
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