Phy 515 class notes for Spring, 1999
Phy 515, Week 2
25 Jan
Uniform convergence of series and the M-test for it. Example of
uniformly convergent series ( ln( 1+x ) in
-b < x < +b ). Also
one that is not: the Fourier series for the square wave and how the sum
of continuous functions converges to a discontinuous function.
Taylor series, formally derived by requiring that the approximating
polynomial have all of its derivatives match that of the given function.
Examples that everyone must know thoroughly:
ex, cos x, sin x, ( 1 + x
)n, ln( 1 + x ).
Show how to write the Taylor series as the exponential of an operator:
ea d / dx f ( x ) = f ( x + a ).
The example of the power series for
1 / ( 1 + x2 ) =
1 - x2 + x4
- x6 + . . .
The function is very smooth for x between minus and plus infinity, but
the series converges only in the interval between minus and plus one.
Why? This can only be answered easily when we look at complex
variables. Essentially, in this case, the function misbehaves when x =
i, and that forces the series to stop converging at a unit distance from
the origin.
27 Jan
Do problem 5-6.6 in detail.
Show an example of a function that has a Taylor series, whose Taylor
series converges everywhere, but whose Taylor series does not equal the
function except at one point:
f( x ) = e -1
/ x2 (for x not equal to zero)
and f( 0 ) = 0.
Taylor's series in several variables. Show how you could figure out the
coefficients the brute-force way by matching coefficients in a formal
expansion. Then show that there is a more efficient way to the solution
by reducing the problem to a problem in one dimension:
Let f( r + t a ) = g( t ),
then do a power series expansion in the single variable t about t = 0.
The result is most concisely stated in terms of the del operator:
f( r + a ) = f( r ) +
( a · — ) f( r ) +
( a · — )2 f(
r ) / 2 + . . .
This is conveniently abreviated as
f( r + a ) =
ea · — f( r )
Along the way, use the chain rule for several variables. What
isthis anyway? Discuss various forms that it can take.
29 Jan
Do in detail the homework problem of computing the monthly payments on a
loan. Show how to do some limiting cases of the final answer,
emphasizing that this is something that should be done every time. The
cases done were N = 1, i = 0, and i small.
Do the pendulum. Set up the torque equation. Re-do it by doing the
energy conservation and differentiating it. Start from energy and
separate variables to get a relation between q
and t. Show how NOT to approximate the answer for small angles by setting
cos q = 1. Next do it for small angles and
show the change of variables to do the integral. Finally examine the
general, non-small-angle case. Rearrange the factors to get the result
into a standard form and define the first elliptic integral.
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