Phy 515 class notes for Spring, 1999

Jan 20 Feb 1 Mar 1 Apr 5 Phy 515 page
Jan 25 Feb 8 Mar 8 Apr 12
Feb 15 Mar 22 Apr 19
Feb 22 Mar 29 Apr 26

Phy 515, Week 3

1 Feb Discuss how to handle powers when you want to keep terms to a consistent order. You sort of have to look ahead to see what they get multiplied by or divided by. That can affect whether you have to keep extra orders or not. In the example of computing the monthly payments on a loan, to keep terms of order "i" you have to expand the ( 1 + i )N in the denominator to second order in i because there is an i to the first power in the numerator. Then when you further expand the powers you have to keep them consistent.
Another example of the practical application of power series is the numerical evaluation of a square root. You can think of the square root as the process of solving the equation

f( x ) = x2 - a = 0.
If you have an approximate answer, such as 1.4 for the square root of 2, then do a power series expansion. Let x0 be the first approximation, and e be the error, then
f( x0 + e ) = 0.
Do a power series expansion of this to get
f( x0 + e ) = f( x0 ) + e f '( x0 ) + . . . = 0
Keep terms only to the first order and solve for e.
e = - f( x0 ) / f ' ( x0 ),
so, x0 + e = x0 - f( x0 ) / f ' ( x0 )
is the next approximation to the root. ( Call it x1 ).
Apply this to the square root and you have
x1 = x0 - ( x02 - a ) / 2 x0 = ( x02 + a ) / 2 x0.
Start with an initial value of x0 = 1.4 for the square root of 2, and the next iteration gives you x1 = 1.4143. You can continue this and essentially double the number of significant figures at each iteration.

Back to the solution of the pendulum for non-small angles. Quickly set the problem up and get to the solution as before. To evaluate the final result, try simply to get the total period of the oscillation. This is four times the time it takes to go 1 / 4 of the way through the cycle, and that allows me to express the period in terms of a definite integral. This is not yet in a standard form, so that added change of variables will have to be done next time.

3 Feb A digression on the notation used for functions. When you try to understand power series and to express them in various notations, it can get confusing. The key stumbling block is that everyone uses the notation f( x ) for a function. It really isn't. If f is a FUNCTION, then f( x ) is the VALUE of the function at the point x. This may seem like a trivial distinction, but when things get confusing this is a solid basis to fall back on.
To solidify the issue further, I went through the precise set-theoretic definition of the words "relation" and "function" in terms of ordered pairs of elements of sets.
On to finishing off the pendulum. Take the result for the period of a pendulum and do the appropriate changes of variable to put it into a standard form. Examine the plausibility of the results. First, the dimensions are correct. Next, look at small angles of oscillation and note that the integral becomes trivial; it also gives the same answer as the standard small oscillation approximation.

5 Feb Discuss some of the homework problems (due date moved to Monday).

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