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Cell["Phys 263 Notebook for 27 May 09", "Subtitle",
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Cell["\<\
The goal again this week is to work through the homework problems, solving \
and plotting as we go.\
\>", "Text",
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Cell["BTM 9.7.8", "Section"],

Cell["The problem we do again and again is:", "Text",
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Cell["\<\
Define our target function on the interval [-1,1]: (Choose one of these at a \
time)\
\>", "Text",
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Cell["First choose the exp() basis functions:", "Text"],

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Cell["Check that we normalized them correctly:", "Text"],

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Cell["And go compute the coefficients:", "Text"],

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Cell["Define the n-th degree approximation as:", "Text",
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Cell["\<\
Now repeat using Sin and Cos.  That is, instead of writing in terms of Exp[2\
\[Pi] n I x/L] with both positive and negative n, we trade for two kinds of \
basis functions, each defined for positive integers n:\
\>", "Text",
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Cell["\<\
Checking that the normalization is right on this interval for a mode or two: \
(change the 3 to something else)\
\>", "Text",
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Note that for n=0 there is no distinct partner -n, and no trade to be made.  \
So we keep \[Phi][0,x] to be the same as it ever was, the constant 1/Sqrt[L]. \
 We can label this mode the n=0 term of the Cos terms, but note we need to \
overwrite the definition for \[Phi]c[0,x]:\
\>", "Text",
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Cell["\<\
Your job: assemble the modes in a g(n,x) function and Plot.
Then repeat, using Cos and Sin for the function Exp[x] on the interval \
(-\[Pi],\[Pi]).
Note that you'll need to modify the modes (and check their normalization) for \
the new interval.\
\>", "Text",
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Cell["\<\
Another example of the utility of Fourier transforms: they are the \"nice\" \
modes for the vibrating string.
Suppose we pluck a string with a certain initial pattern of displacements, \
and for our first pass let's
assume we start everything from rest (i.e. our particular solution will be a \
sum of Cos[\[Omega] t]'s with no Sin's).\
\>", "Text",
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Cell["Define a starting shape for the string:", "Text"],

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Cell["Fourier decompose in terms of sin():", "Text"],

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Cell["\<\
Don't forget to check that is is OK:  (actually this one isn't--- I'm asking \
you to fix it by changing the line above until the line below is coming out \
to its proper value of 1)\
\>", "Text",
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Cell["Then assemble the modes and plot:", "Text",
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Cell["\<\
Once we have the coefficients (as determined from t=0 data) we can use them \
to form the solution at general t:\
\>", "Text",
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Cell["And of course, Animate:", "Text",
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Cell["\<\
If the function takes some time to compute, you can make a better movie by \
computing the frames once and for all:\
\>", "Text",
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 "Your job: repeat for the function of problem 9.7.13, where the string \
length is L=\[Pi] and the initial displacement is:\n\"between 0 and \[Pi]/4, \
rises linearly from 0 to 1; from \[Pi]/4 to 3\[Pi]/4 falls linearly from 1 to \
-1; between 3\[Pi]/4 and \[Pi], rises from -1 back to 0\".  Again, you'll \
need to change the limits of integration and the normalization of the modes \
above.\nAnd a piecewise continuous function can be written using ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 "'s Piecewise construct.\nE.g. here is the triangle wave from above:\n"
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Cell["\<\
Challenge: change the boundary conditions so we start with some velocities.  \
How should the velocities be chosen if you want to launch a wave in one \
direction only?\
\>", "Text",
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Cell["Complex Functions", "Section",
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Cell["\<\
The second theme of the week is thinking about complex valued functions of a \
complex variable.\
\>", "Text",
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Cell["BTM 6.1.5", "Subsection",
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Cell["\<\
... asks about analyticity and the Cauchy-Riemann equations.
The starting idea is that in general a mapping of complex to complex does not \
have a good derivative,
if by derivative you mean take the limit as \[Epsilon] goes to zero of (f[z+\
\[Epsilon]]-f[z])/\[Epsilon].  The hang up is that \[Epsilon] is now
understood to be complex, implying there are at least two directions we can \
approach zero, real and imaginary,
and these two will not agree in general.
Try it out, writing in x,y coordinates:  here is a random pair of functions: \
(feel free to change)\
\>", "Text",
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Cell["\<\
Your job: repeat for the u and v given in the problem, verifying that the \
real and imaginary parts of the derivatives agree, i.e. that the \
Cauchy-Riemann relations are obeyed.\
\>", "Text",
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Cell["To visualize complex functions, here is a nifty package:", "Text",
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Cell["which defines the following coloring of th ecomplex plane:", "Text",
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Cell["\<\
You give in any complex valued function expressed in terms of x and y, and \
f(x+iy) is plotted in color:
the intensity encodes the magnitude and the phase is encoded in the color \
wheel (with red for real of course).\
\>", "Text",
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Cell["\<\
Here is the function z^2, which has a double zero at z=0.  Note how we swirl \
around the color wheel twice.\
\>", "Text",
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Cell["\<\
And here is 1/z^3 which has a \"triple pole\" singularity at z=0.  The wheel \
swirls three times the other way.\
\>", "Text",
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Cell["Some functions are polynomials, which have roots:", "Text",
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Cell["These zeroes show up as black dots in:", "Text",
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Cell["\<\
Or sometimes there are poles at the zeroes of some denominator:\
\>", "Text",
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Cell["\<\
Your job: repeat for the functions of BTM 6.17: f(z) = 1/(1+z^4) and f(z) = \
1/(z^4+2z^2+1).  Also, you might plot
the function of BTM 6.1.5 given above as u+Iv.  Can you express the function \
in simple terms using z=x+Iy?\
\>", "Text",
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Cell["\<\
And by the way, not all functions are polynomials.  Here is good old Sin and \
its zeroes.\
\>", "Text",
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Cell["\<\
Your job: prove that all the zeroes of sin lie on the real axis, i.e. there \
are no new funny ones out in the complex plane.  Along the way, notice \
various relationships, such as Sin[i x] = i Sinh[x]\
\>", "Text",
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Cell["\<\
Next, visualize the n'th roots of unity, i.e. all the numbers z such that \
z^n=1.
In particular, Plot f(z) = z^6-1 and look for the zeroes.\
\>", "Text",
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Cell["Then Solve for the zeros and add them up:", "Text",
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Cell["\<\
Your job on the paper homework: write these roots in polar and then x,y form \
before you add them up.
But here ...\
\>", "Text",
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Cell[TextData[{
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 " can find at least one possibility:"
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Cell["\<\
But are there others? I.e. QPlot the function Exp[z]+2 and look for any black \
dots.\
\>", "Text",
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Cell["\<\
Here is a random 4th degree polynomial with 4 roots (two conjugate pairs)\
\>", "Text",
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Cell["\<\
BTM 6.2.14 asks about these polynomials: f(z) = z^4 + 2z^2 + 1 and f(z) = z^6 \
+ 2 z^3 + 1.
How many roots do these ones have?  And are they simple zeroes, or perhaps \
double zeroes?
How can you tell from the QPlot?\
\>", "Text",
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