Complex Analysis

Math 312 Spring 1998  MWF 10:30am - 11:25am
Buckmire  Fowler 112

Class �7 (Wednesday January 28)

SUMMARY Limits of Complex Functions and the Point At In�nity CURRENT READING Brown & Curchill, pages 33-40 NEXT READING Brown & Curchill, pages 40-45

Update on Class �5 Yes, closed sets can be connected. Open connected sets are more interesting because they are also called domains or open regions. If a set is closed and connected it’s called a closed region.

If a set does not have any limit points, such as the set consisting of the point {0}, then it is closed. [It contains all its limit points (it just doesn’t have any limit points).]

Remember, if a set contains all its boundary points (marked by solid line), it is closed. If a set contains none of its boundary points (marked by dashed line), it is open.

Also, sets can be both open and closed. An example is the set C (the Complex Plane). It has no boundary points. Thus C is closed since it contains all of its boundary points (doesn’t have any) and C is open since it doesn’t contain any of its boundary points (doesn’t have any).

Also, sets can be neither open or closed. The set 0 < |z| ≤ 1 has two boundaries (the set |z| = 1 and the point z = 0). It contains the �rst boundary (|z| = 1), so it is not open, but it does not contain the boundary point z = 0 so it is not closed. z = 0 is also a limit point for this set which is not in the set, so this is another reason the set is not closed.

Limits

Suppose that f(z) is de�ned on a deleted neighborhood of z_{0}. In order to say that lim_{z→z_{0}} f(z) = w_{0} we must be able to show that

∀ϵ > 0, ∃δ > 0 ∋ |z − z_{0}| < δ ⇒ |f(z) − w_{0}| < ϵ

This may look like dense mathematical language, but in english this means that for every positive number ϵ (no matter how small) there exists a number δ (which depends on the choice of ϵ) so that regardless of how close you get to the point z_{0} in the deleted neighborhood around it in the z-plane you can also get arbitrarily close to the value w_{0} in the w-plane.

Rules on Limits The rules on limits of complex functions are identical to the rules for limits of real functions of real variables (as you’d expect) Suppose that lim_{z→z_{0}} f(z) = w_{0} and lim_{z→z_{0}} F (z) = W_{0} then

lim_{z→z_{0}}[f(z) + F (z)] = w_{0} + W_{0}
lim_{z→z0}[f(z)F (z)] = w_{0}W_{0}
lim_{z→z0} \frac{f(z)}{F (z)} = \frac{w_{0}}{W_{0}}  (W_{0} ≠ 0)
lim_{z→z0} |f(z)| = |w_{0}|
lim_{z→z0} c = c
lim_{z→z0} z^{n} = z_{0}^{n}
lim_{z→z0} P (z) = P (z_{0})  (where P (z) is a polynomial )
IF f(z) = u(x, y) + iv(x, y),  z_{0} = x_{0} + iy_{0}  and w_{0} = u_{0} + iv_{0} THEN

lim_{z→z0}f(z) = w_{0} if and only if lim_{(x,y)→(x0,y0)}u(x, y) = u_{0} and lim_{(x,y)→(x0,y0)}v(x, y) = v_{0}

\underline{Examples}

(a) lim_{z→1+2i}2|z| + iz^{2} + 2.5 − .1i =

(b) lim_{z→3πi}ze^{z} =

(c) lim_{z→0} \frac{z^{8} + z^{4} + z^{2} + z − 1}{z^{3} + 4z^{3} − 9} =

(d) lim_{z→1−i}2xy − ix^{2} − iy^{2} =

Point at In�nity

When dealing with real numbers we often speak of two different concepts, denoted − ∞ and + ∞. However, the complex number in�nity is represented as one particular point in the Argand plane. We call this the point of in�nity and we rename the Argand plane the extended z plane or the extended complex plane when we include it. The point at in�nity can be considered to be the image of the origin z = 0 under the mapping w = 1/z To compute complex limits involving ∞ we use this idea:

lim_{z→∞} f(z) = lim_{w→0}f (\frac{1}{w} )

In order for us to say a complex function f(z) becomes unbounded at a point z_{0}, (i.e. f(z_{0}) = ∞) we must have that

∀M > 0, ∃δ > 0 ∋ 0 < |z − z_{0}| < δ ⇒ |f(z)| > M

\underline{Examples}

(a) lim_{z→∞} \frac{2z + 3}{3z − i} =

(b) lim_{z→∞} z^{2} + 3z − 5i + 4 =