8.3 Functions of a Complex Variable
8.3.4 Powers
First of all, as before, we consider the powers zn with natural exponents n∈N: w=zn= (x+ iy)n=|z|n(cosϕ+ i sinϕ)n=|z|neinϕ =|z|n(cosnϕ+ i sinnϕ),
where at the end we have used the Euler formula and thus written down its extension:
the
Moivre formula: zn=|z|n(cosnϕ+ i sinnϕ).
This implies for the
absolute value of the n-th power: |zn|=|z|n
and for the
argument of the n-th power: arg(zn) = narg(z).
We want to discuss two examples in more detail:
1) As the first example we choose the quadratic function, i.e. n= 2 :
quadratic function: w=u+ iv =z2 = (x+ iy)2 = (x2−y2) + i2xy=|z|2e2iϕ, meaning for the real part: u=x2−y2 and for the imaginary part: v = 2xy, respectively for the absolute value: |w|=|z|2 and for the argument: arg(w) = 2 arg(z).
Firstly we determinesome image points:
w(±1) = (±1)2 = 1,
w(±i) = (±i)2 =e±iπ22 =e±iπ =−1 and w(1±i) = (1±i)2 = (√
2e±iπ4)2 =±2i.
Then we look at thevertical straight linex= 1 :u=x2−y2 = 1−y2andv = 2xy= 2y.
Thus 1−u=y2 = v42, i.e. u=−v42 + 1, which is the equation of a parabola open to the left side.
Analogously you can show that thehorizontal straight liney= 1 becomes the parabola open to the right sideu= v42 −1.
Apparently the unit circle |z| = 1 is mapped onto itself by the quadratic function:
|w|= 1.
From u = x2 − y2 = const. you can see that the hyperbolae with the bisectors as asymptotes become vertical straight lines, and from v = 2xy = const., that the hyperbolae with the axes as asymptotesbecome horizontal straight lines.
The following figure gives an overview over the whole mapping. Note that the left-side half of the pre-image plane is omitted, since the image of the right half alone covers the entire w-plane.
Figure 8.9: Right half of the z-plane and the entire (upper) w-plane for the quadratic function
Insert: Rubber sheet: You can imagine the mapping procedure approximately in the following way: think of the right-half of the z-plane as made of an elastic sheet and then rotate the positive and negative halves of the imaginary axis in opposite direction by 90◦ around the origin until they meet each other along the negative real axis.
The image of the left half of the Gauss z-plane leads to a second cover of the whole w- plane. We already encountered part of this in connection with therealquadratic function, where the image of the negative pre-image half-axis covered the positive image half-axis a second time, so that the square root function could only be defined over the positive half
line. In order to allow now for the complex quadratic function an inverse function over the whole plane, mathematicians cut the two picture planes, imagined lying one over the other, (the cut being e.g. along the negative real axis), and connect the upper edge of the cut of the upper sheet with the lower edge of the cut in the lower sheet and think of the lower edge of the upper sheet as penetrating “straight through the other connection”, stuck together with the upper edge of the lower sheet. The whole construction consisting of the two planes connected crosswise along the negative real axis is called aRiemannian surface with two sheets. Thus we can say: The complex quadratic function maps the z-plane bi-uniquely onto the two-sheeted Riemannian surface, where the special position of the cut is arbitrary. Decisive is, that the cut runs between the two branching points 0 and ∞. The next figure tries to visualize this situation.
Figure 8.10: Riemannian surface of the quadratic function
During the motion of a mass point e.g. on the unit circle in the z-plane with the starting point z = 1 the image point w runs along the unit circle in the upper w-plane, however, with twice the velocity until it dives near z = i, i.e. w= −1 into the lower sheet of the Riemannian w-sheet. It goes on running on the unit circle in the lower sheet, reaches for z = −1 the point w = +1 in the lower sheet and appears only for z =−i at the diving pointw=−1 again in the upper sheet, to reach finally on the upper unit circle forz = 1 the starting point w= 1.
2) A similar construction holds for the cubic function withn = 3 : cubic function: w=z3 =|z|3e3iϕ =|z|3(cos 3ϕ+ i sin 3ϕ),
meaning for the absolute value: |z3|=|z|3 and for the argument: arg(z3) = 3 arg(z).
We determine only a few image points:
w(±1) = (±1)3 =±1, w(i) = i3 =e3πi2 =−i and w(1 + i) = (1 + i)3 =−2(1−i).
We see that already one third of the z-plane is mapped onto the entire w-plane, and that the entire pre-image plane is mapped onto a Riemannian surface consisting of three sheets which are cut between 0 and ∞ connected with each other. The following figure sketches the situation:
Figure 8.11: One third of the z-plane and the upper sheet of the w-plane for w=z3 Continuing in this manner an overview over allpower functionsw=zncan be reached.
In particular, one n-th of the z-plane is bi-uniquely mapped onto the whole w-plane or the whole z-plane onto a n-sheeted Riemannian surface. At least in principle this gives a feeling for the mapping action of complexpolynomials: Pm(z) =
m
P
n=0
anzn.
For every complex polynomial of m-th degree the Fundamental Theorem of Alge- bra guaranties the existence of m complex numbers zn, such that the sum can be represented as a product of m factors:
Pm(z) =
m
X
n=0
anzn =am(z−z1)(z−z2)(z−z3). . .(z−zm−1)(z−zm) :
Fundamental Theorem of Algebra:
∃zn ∈C, n = 1,2,3, . . . , m: Pm(z) =
m
P
n=0
anzn=am
m
Q
n=1
(z−zn).
Exercise 8.8 Concerning the fundamental theorem of algebra:
Show with help of the Fundamental Theorem of Algebra, that the sum respectively the product of the m zero points wn of a polynomial Pm(w) = 0 holds:
m
P
n=1
wn = −am−1
am
respectively
m
Q
n=1
wn= (−1)ma0 am.
For the complex infinite power series
∞
P
n=0
an(z −z0)n, the ones the mathematicians call also “analytic functions”, we report without proof that all these series converge absolutely inside a circle domain|z−z0|< Rwith the radiusRaround the development centrez0 and diverge outside that region. Only now can we really understand the “convergenceradius”
R, that can be determined according to the criteria of convergence we have explained earlier. For instance for the complex geometric series
∞
P
n=0
zn = 1−z1 the singularity atz = 1 restricts the radius of convergence toR = 1, as we have seen in Section 6.5 with help of the quotient criterion. Now we want to examine in more detail three very important power series as examples: the natural exponential function, which we already met, and the complex sine and cosine functions.