blob implies maximum modulus principle. harmonic function is locally the real part of an analytic function, so get maximum principle
Another point of view
If sqrt(z) has a removable singularity at 0, then we have an analytic
function f(z) defined in a disc centered at 0 so that
f(z)^{2}=z. But then we can differentiate, and get
2f(z)f´(z)=1 for z<1. What if z=0? Well, as z>0, I think
that sqrt(z)>0, so f(0)=0. But:
2·0·SOMETHING=1. This can't be true. So where is
the contradiction?
Isolated singularity?
sqrt(z) does not have an isolated singularity at 0. There is no r>0
so that sqrt(z) is analytic in all of 0<z<r.In fact, if you
look at sqrt(z) as it travels "around" a circle, you will see that
when you get back to where you started, the arguments do not match up
(onehalf of 2Pi is not the same as 0, mod 2Pi). So there is a need to
be a bit careful.
A Laurent series from the textbook
z/[sin(z)]^{2}.
Is residue additive?
Is residue multiplicative?
A residue e^{z+(1/z)}
A version of the grownup Residue Theorem
A real integral
Goals
Exam?
Examples
The child's residue theorem
A return to the snowball
A few lectures ago we analyzed a problem about the melting of a
snowball. I felt somewhat dissatisfied with the statement of the problem, because
I didn't think the statement gave enough background. Let's look again at the problem.
The setup is certainly simplified from "real life". The snowball is a sphere of radius r,
with surface area S=4Pi r^{2} and with volume V=[4/3]Pi r^{3}.
Put the snowball in a warm environment. The snowball will melt, of course. But how
does it melt? If you think about it, the volume, V, will decrease, but more precisely
it will decrease as the snowball absorbs heat. Heat is absorbed through the surface of
the snowball. (I don't think in this model that we should imagine a little machine
in the middle of the snowball radiating heat!) So I think that the rate of change of
the snowball's volume should be directly proportional to the surface area. That is,
there is a constant k so that dV/dt=kS. What happens to the radius of the snowball?
Well, since V=[4/3]Pi r^{3}, then (r varies!)
dV/dt=[4/3]Pi 3r^{2}[dr/dt]. Match this with dV/dt=kS, and, wow!,
we see that dr/dt=k. So if we believe this model, then (as the original problem
statement specifies) the radius of the snowball is constantly decreasing.
The surface area of a baby?
The snowball problem is quite relevant to certain aspects of biology. There are
approximate formulas for the surface area of a baby. Such formulas can be useful
if estimations of fluid balance (sweat) or temperature change are needed. Babies
are smaller than adults, and such balances may be very unstable. If the balances
are not maintained, illness and even death can result.
So how does the radius change?
What is (7.3)^{2}?
What is (50)^{2}?
The chip company
Biological systems are complicated
Prozac etc. Animal tranquilizers.
The formula and a picture
But the ERROR
QotD
What is the approximate value of (7.98)^{1/3}? here I asked that people
not use calculators, and use the linear approximation scheme discussed above
Here's the whole story
Multiple names: linear approximation, differfential, etc.
Bob Roundy
Having little better to do this morning while at the car dealer's waiting room (car
being serviced) I looked at the recent SI swimsuit issue, and of course paid the most
attention to the masthead, and there you were, a MANAGER. Have you been a manager for
long? I am confused. But it was good to see you there. I think it took about 45 to 50
minutes driving from Douglass to Busch yesterday afternoon, which at that time usually
takes about 5 to 7 minutes. Driving was yucky.
Order of a zero
I should have mentioned, darn it, that there is an alternate
characterization of the order of a zero. If f(z_{0})=0, then
either f(z) is always 0 everywhere, or if we look at the power series
for f(z) centered at z_{0}, it will look lik
SUM_{n=0}^{infinity}[f^{(n)}(z_{0})/n!](zz_{0})^{n}.
But if f does have a zero of order k, the first k terms (starting
counting from the n=0 term!) are 0, so in fact, "locally"
f(z)=SUM_{n=k}^{infinity}[f^{(n)}(z_{0})/n!](zz_{0})^{n}
with f^{(k)}(z_{0}) not 0. I can factor out
(zz_{0})^{k}, and the power series will still
converge. Let me call the sum g(z). Since g(z) is the sum of a
convergent power series, g(z) is an analytic function, and
g(z_{0}) is not 0 (because it is actually
f^{(k)}
THIS IS NOT COMPLEX
ANALYSIS SO YOU CAN DISREGARD
IT! You will never, never, never, see such a function in a complex analysis course. Our functions are either 0 everywhere or they are zero at isolated points. The picture was produced using these commands: f:=(x,y)>piecewise(x<0,sqrt(x^2+y^2),x<1,abs(y),sqrt((x1)^2+y^2)); plot3d(f(x,y),x=3..3,y=3..3,grid=[50,50]);

Isolated singularities
Examples
If f(z) is not the zero function, and if f(z_{0})=0, then the
function
Names
Properties
Reasons
the pow
RRST
POLE
CW
2.4 20 24 do any 1 of these!
2.5 1 6 7 10 12 14 22b
2.6