AMAT 315

MWF 1:40-2:35 ES 147

Instructor: Professor Edward
Thomas ES 132F

Phone 442-4623

Best way to reach me is by
email: et392@albany.edu

Office hours: MWF 10:25-10:45,
12:50-1:30, 2:45-3:15 and by appointment

A link to the class webpage
will be found at : http://www.albany.edu/~et392/

Text: Mathematical Methods in
the Physical Sciences by Mary Boas. *You need
to have this text on the first day of class, not at some indeterminate later
date. If you don’t have the text, I will take this as a sign of LACK OF
PREPARATION.*

* Same ground rules as usual: *

*> First...there will be NO CELL PHONES, LAPTOPS or any other
type of electronic devices in use during class. Please take care of
business and TURN THEM OFF before you enter the classroom.*

*>Second…please DO NOT **COME TO CLASS LATE** **as it is disruptive. Be in your seat, mentally
alert and ready to participate, at 1:40 when class begins.*

*> Third …If you get sick or have some other
kind of emergency, please
get in touch with me as soon as you can so we can work things out.*

* > Fourth, classes begin on Wednesday, January 22nd ( You wouldn’t believe it but in the past some peeps thought
they could begin classes on a day of their own choosing. That was a BIG
mistake.)*

The syllabus for this course was developed
in conjunction with the Physics Department. The emphasis is on those aspects of
Mathematics that are can be readily applied to problems in Physics and
Engineering. This second semester, we
will concentrate on some of the classical Partial Differential Equations of
Physics and then take up Complex Analysis and its applications to Applied
Mathematics.

There will be assignments every day. These
will be corrected and returned to you at the next class meeting.

Assignments will count as one third of the
final grade, the other two thirds coming from the mid-term and final exams.

Assignments:

1) Page 620 # 2 (a)

2) On Friday, as we went about solving the steady state
heat equation, we came to a fork in the road. Suppose we opted to choose the
separation constant to be positive : c = k^{2}.
Show that this would lead to the trivial solution. (As a matter of fact, show that
when you figure out what X(x) and Y(y) are, the right
and left boundary conditions imply that X(x) must vanish identically.)

3) a) Using the first 3 terms of the Fourier series
developed in class, estimate T(5, y) for y = .1, .2 and .5 ( don’t forget the 400/Pi
out in front)

b)
For future reference, figure out the general form of cos(
nPi/2) ( you don’t need a single formula here, just
the three different values depending on what n is)

4) a) Continuing with the last homework problem, make better
estimates by averaging the 3-term partial sums ( which you already computed)
and the 4-term partial sums, as discussed in class.

b) Compute the Fourier sine
coefficients…the bn’s…for the function f(x) = 96 when
0<x<L/2, f(x) = 0 when L/2<x<L.

5) a) A curious feature of Fourier
series…continuing part (b) above, and using the series we got in class,
estimate the value of T (L/2,0) by averaging the 7^{th} and 8^{th}
partial sums.

(b) ( The wrong
prong) Referring to the heat diffusion equation, what goes wrong if you take a
positive separation constant c = k^{2}?Why won’t your solution make
physical sense?

(c) Expand f(x) = ( 100/L)x
in a Fourier sine series on the interval 0<x<L, as set up in class.

6)Turning to the 1-dimensional Schrodinger equation with V=0,
as we did in class, first separate variables and call the separation constant
E. Next, find the solution for T(t), not forgetting
the properties of the imaginary constant. (The answer is in the book).

Now let k^{2}= 2mE/h^{2} ( h = normalized Plank’s constant) and show that, if we
assume psi(x) =0 when x=0 and L, then psi is a pure sine term and k = n Pi/L.

With these assumptions, this is called the “particle in a box” problem
of quantum mechanics.

We’ll pick it
up there next time.

7) Page 633 #11. (Don’t bother
to plot Psi^2)

8) Explain the reasoning why
we select a sine or cosine in the time-dependent part
of the eigenfunctions
in “plucked” versus “ hammered”.

And, do problem
#1 on page 637 Added Tuesday at 1;30…this freakin’ problem took me three pages to write out. In the
end, you get a whole bunch of stuff to cancel out. In case I didn’t mention it,
the first two terms of the answer are given as ( 4.9)
on page 635.Kudos to anyone who actually is able to stay with it all the way.

9) page 590 # 4
Verify this by writing out the first 5 terms of each series …then if you want,
you can do the general proof using the sigma notation ( pain in the neck)

10) On page 646, extend the figure to
include the next two columns and rows. So all together, there will be sixteen
modes of vibration, including the four that are already pictured.

Then, on a lighter note, draw a 3-dimensional picture of the mode n=0,
m=2. It should resemble a familiar object from south of the border.

11) page 51 write # 2- 8 in
polar form and page 52, do # 1-7

The Midterm Exam
will be on Friday, March 14^{th}^{}

^{ }

^{12) page 63 # 14, 18-21 and
page 66 # 3, 4, 5, 9 }

^{13) page 667 # 1-4, 6 and 9}

^{14) Part 1.
Take # 1-4, 6, 9, 14 and 15 on page 667 and test each using the CR equations. Which
are differentiable and which are not?}

^{Part2. Assume that f(z) is
differentiable. Compute what you get for the derivative of f(z)
when you let z + delta(z) approach z along a vertical line. Equate this to the
expression we got in class ( using a horizontal line)
and show that you get two equations and they are none other than the CR
equations !!}

^{15) page 673 #45 this problem
establishes an intriguing connection between the Cauchy Riemann equations for a
function f(z) = u +iv and the physics of the associated vector field F = vi + uj where the red color denotes the standard unit vectors
from Calc III.}

^{16) First, prove part 1 of the Beautiful Theorem,
namely: If f = u +iv is analytic in a region R, then both u and v are harmonic
in R. (Hint: Express the second partials of u with respect to x and y as mixed
partials of v using the CR equations….add ‘em up to
get 0. do the same thing for v.)}

^{ Second, in
the example we did in class to illustrate part 2 of the Beautiful Theorem,
figure out what the final function f(z) turns out to
be. }

17) page 676 # 1.
Do this using the “ordinary calculus way”, using the antiderivative
for f(z) = z and verify that you get the same answer
as we did by parametrizing. (b) do
problem 3(b). Parametrize the semicircular arc using
the polar angle , theta, and parametrize
the straight line from -1 to +1 just like we did in class with problem 1.

18 ) Let C be the unit circle centered at the origin and
oriented counterclockwise. Calculate the integral of f(z) = 1/z and also f(z) = 1/z^{2}
around C , *using the polar angle theta as
parameter. * (The point of this is to contrast your
answers with Cauchy’s Theorem.)

19) Show that co(z)
and sin(z) satisfy everybody’s favorite trig identity. Then do problems # 3, 6,
9 on page 69 and # 1-6 on page 74)

20) page 677 # 17, 18, 19 Note: to apply Cauchy’s Integral
Formula, the denominator must have the form z-a. Plus, compute the integral of
z dz/ (z^{2} – 2) around the circle of radius
1 centered at z =1

21) page 678 # 22, 23 plus
compute the integral around the circle of radius 2 centered at the origin of e^{z} divided by z^{2} + 6z + 5

22) A handout on Laurent series…copies
posted on my door.

23) Find the Laurent series expansion of f(z) =1/z^{2}( 2 – 3z) which is valid in a punctured
disk

0< mod(z)<
? and circle
the residue.

24) (a) Expand f(z)
= 1/ z^{5}( 2-z)(5-z) in the innermost region (centered at z_{0}
= 0) and find the residue. (b) Expand g(z) = 1/(2-z)^{3}(3-z)
in the punctured disk at z_{0}=2 and find the residue.

25) Find the integral of dz/z(4-z) around the following circles. Find residues by
Laurent series when needed: (1)the circle centered at
0 of radius 1, (2) center 2, radius 1 and (3) center 3 and radius 2. Plus page 686 # 14 and 15(
Sometimes in a Laurent series, you need to expand 1/(1+r) rather than
1/(1-r). Well. 1/(1+r) expands as the __alternating__
geometric series 1 – r +r^{2} – r^{3} + etc

26) (a) Prove that if f(z)
= 1/(az+b)(cz+d) then the
residues at the two singularities of f are the
negatives of each other. (b) Compute the integral around the unit circle of 1/i times (2z^{2} + 5z + 2)^{-1} and (c)
compute the integral of e^{iz} (1+z^{2})^{-1
}around a simple closed curve C that lies in the upper half plane and
encloses the complex number i.

27)
page 699 # 1, 2, 5 Hint on #1: z^{2 }+ 26iz -5 factors to give two
singularities ( poles) and one of these
lies inside the unit circle.

28)
pg 699 #10. Be sure, at the very least, to pay lip
service to what happens to the integral along C^{+}_{R }as R
approaches infinity.

29) page
699 # 13 and 15 ( if you don’t get the author’s answer on #15, join the club)

30) (a)Water
flows in and out of a wedge shaped channel bounded by the rays theta = 0
degrees and theta = 60 degrees. Find the equation of a typical streamline. (b)
Suppose a particle starts at the point ( 1,1) and
flows along its streamline ultimately heading southeast. Approximately where
will it cross the line x = 3? (c) Discuss what other wedge shapes can be *readily** *analyzed this way.

31) Prove that e^{z}^{
}and log(z) satisfy the Cauchy-Riemann equations.
There was a handout reminding you of the real and imaginary parts of these two
functions. In particular, log(z) = ln((x^{2}
+ y^{2 })^{1/2}) + i(arctan (y/x)).

32) Several problems on transforming rectangles into
annular regions and vice versa with e^{z} and
log(z). Plus page 716 #4