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Disclaimer
Use these notes at your own risk. This copy of my transcribed notes is here for
your convenience, but neither the instructor nor I will be held responsible for any
mistakes contained therein. In particular, this isn’t an “officially sanctioned”
transcript of the class notes.
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Symmetry around the circle
Figure 1: The point Q is symmetric to the point P with respect to the circle.
We discuss points that are symmetric with respect to a circle. Consider the
unit circle, and the transformation z → 1/z. This is called inversion. Any pair
of nonintersecting circles can be taken by some inversion to a pair of cocentric
circles.
Figure 2: The black circles form a Steiner chain of the red and blue circles.
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Proposition 2.1 (Steiner’s Theorem). Consider the construction in Figure 2.
The black circles are known as a Steiner chain of the red circle and the blue
circle. Prove that the existence of the Steiner chain depends only on the the red
and blue circles (and not, say on where we place the first black circle).
Proof. Invert. You either get cocentric circles or you don’t
Exercise: image of a point under inversion can be found by compass alone.
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Stereographic Projection
Points (x1 , x2 , x3 ) on the sphere go to (x1 + ix2 )/(1 − x3 ) on the complex plane.
How do we express points (x1 , x2 , x3 ) on the sphere in terms of z on the plane?
Taking x3 constant, this projects to a circle on the complex plane centered at
0. Thus
x21 + x22
(1 − x3 )2
1 − x23
=
(1 − x3 )2
1 + x3
=
1 − x3
|z|2 =
an so x3 =
|z|2 −1
|z|2 +1 .
Thus
2
+1
z − z̄
x2 =
i(1 + |z|2 )
z + z̄
x1 =
(1 + |z|2 )
x3 =
|z|2
The equation of a circle on the sphere is A1 x1 + A2 x2 + A3 x3 (intersecting
the sphere with the plane). Substituting this into our equations of x1 , x2 , x3 in
terms of z, we find α|z|2 + βz + βz + γ = 0, which is a circle (or a line).
Home reading: distance between points. There is an interesting relationship
between (Euclidean) distance between two points on the sphere and the distance
between two points on the Complex plane.
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Complex Differentiability
Consider the real function xk sin(1/x). It is differentiable, but not infinitely
many times. Consider the function f (x) = exp(−1/x2 ) when x > 0, and 0 when
x ≤ 0. f 0 (0) = exp(−1/x2 )/x = 0. In fact, all derivatives are 0. Clearly, the
function is not zero in any neighborhood of 0. Is there a infinitely differentiable
function whose taylor series does not converge at all? Yes!
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Theorem 4.1 (Borel’s Theorem). For all {an }, ∃C ∞ function f such that
f (n) (0) = an for any sequence an .
All these phenomena do not occur in complex differentiation. If a complex
function is differentiable once, it is differentiable infinitely many times, and the
power series are well-behaved.
Definition 1 (Holomorphic). A function f is holomorphic in a domain Ω ⊂ C
if for any z ∈ Ω the limit
f (z + h) − f (z)
h→0
h
f 0 (z) = lim
exists, for h ∈ C.
Take f (z) = u(x, y) + iv(x, y). We can thus consider f as a map from R2 to
R2 . Consider the Jacobian matrix


∂u ∂u
 ∂x ∂y 
 ∂v ∂v 
∂x ∂y
We must have
∂u ∂v
=
∂x ∂y
∂v
∂u
=−
∂y
∂x
(1)
(2)
(3)
These are the Cauchy-Riemann equations. These are derived from taking the
derivative of f from the direction of the real axis, and then taking the derivative
from the direction of the complex axis (If f is holomorphic, the derivative taken
from any direction must be the same).
Excercise: if the function f is continuous and satisfy the Cauchy-Riemann
equations, then f is homomorphic. What real functions can be the real part of a
holomorphic function? Can x2 + y 2 be the real part of a holomorphic function?
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