1.4 • THE GEOMETRY OF COMPLEX NUMBERS, CONTINUED 2 5In this case, Arg( 4i) = ~ and arg( 4i) = { ~ + 2mr : n is an integer}.
As you will see in Chapter 2, Arg z is a discontinuous function of z because
it "j umps" by an amount of 271" as z crosses the negative real axis.
In Chapter 5 we define e• for any complex number z. You will see that this
complex exponential has all the properties of real exponentials that you studied
in earlier mathematics courses. That is, e'>e•2 = ez^1 +•2, and so on. You will
also see, amazingly, that if z = x + iy, then
e" = e"+iY = e"(cos y + isin y). (1-31)
We will establish this result rigorously in Chapter 5, but there is a plausible
explanation we can give now. If e• bas the normal properties of an exponential, it
must be that ex+iy = e"'eiY. Now, recall from calculus the values of three infinite
series: e"' = I::^00 tr xk, cos x = I::^00 (-!))f2ny "" x^2 n, and sin x = I::^00 ~ '-"" x^2 n+l. Sub-
k=O n=O n=O
00 00
stituting iy for x in the infinite series fore"' gives eiY = I:: fa (iy)k = I:: f,_ikyk.
k=O k= O
At this point, our argument loses rigor because we have not talked about infinite
series of complex numbers, let alone whether such series converge. Nevertheless,
if we merely take the last series as a formal expression and split it into two series
according to whether the index k is even (k = 2n) or odd (k = 2n + 1), we get
ei11 = '°' ..!_ikyk + '°' ..!_ikyk
L..J k!. L..J k!
k is even k is odd
00 1 00 1
= L --i2ny2n. + L i2n+Iy2n+l
n = O (2n)! n=O (2n + 1)!
= f:-1-{i2f y2" + f: 1 (i2t iy2n+1
n=O (2n)! n=O (2n + 1)!
00 00
_ '°' 1 ( )" 2n · '°' 1 ( )n 2n.+ l
-~(2n)! -l y +i~(2n+l)! - l Y= cosy+isiny.Thus, it seems the only possible value for e is that given by Equation (1-31).
We will use this result freely from now on and, as stated, supply a rigorous proof
in Chapter 5.
If we set x = 0 and let 8 take t he role of yin Equation (1-31), we get a
famous result known as Euler's formula: