ChapterZero

Use Cauchy’s integral formula to calculate the integral of on the counterclockwise unit circle about the origin. Not so bad. Use it to calculate the integral of ; a little trickier and a lot messier, but not daunting. Now, use it to calculate the integral of along the same path.

Multivalued functions have been giving me headaches for the past couple of days. Identifying branch points, and picking branch cuts (not to mention explicitly spelling out the formulas for branches of nontrivial functions) seems to be something of a black art. It’s moments like these when I wish I took an undergraduate complex analysis course.

That said, where’s the error in the following reasoning:

Here’s ‘an extremely far-reaching criterion’ that can be used to determine the convergence of a series (a proof can be found in Knopp’s Theory and Applications of Infinite Series):

A series of complex terms, for which

with bounded– where is complex and arbitrary, and – is absolutely convergent if, and only if,

For the series is invariably divergent. If , both the series

are convergent.

Furthermore, if , the series is divergent.

In itself, this is a useful result, but it is particularly useful for determining the convergence properties of a power series on its circle of convergence:

If, as in the preceding theorem,

where is arbitrary, , and is bounded, the series is absolutely convergent for , divergent for every , and for the points of the circumference , the series will

1. converge absolutely, if
2. converge conditionally, if except for where it diverges
3. diverge, if

That’s a pretty powerful corollary. As an example, use it to show that the generalized harmonic series (my term for the series ) diverges only at one point on the unit circle!

What are some general techniques for showing that a power series expansion of a function converges to that function? As an important example, if you’re dealing with a real-valued function, what techniques exist for showing it is real-analytic, other than dealing directly with epsilon-delta limit arguments?

The only power series equality that I can remember proving directly is when . In real analysis, I can’t remember ever proving a function is real-analytic (not that we didn’t ever do it, maybe we did), and in complex analysis, we use Cauchy-Reimann to show that a function is analytic, so equal to its power series.

Lately I’ve been visiting conservative news sites, to “get into the minds of the enemies”, as it were. Tonight I happened across an article at townhall.com on the efficacy of advocating condom use vs. abstinence only in sex ed courses.

Abstinence only education is polemic because some expect and intend for sex education to be normative, while others intend for it to be informational. In the latter camp are those who, like myself, believe sexual education is to be exactly that: education, not sermonizing.
Call me a pie in the sky optimist– or maybe I was just an exceptional teenager–, but I tend to believe giving a teenager the information to act responsibly is sufficient. When I took sex ed, the course was purely informational.

The former camp is occupied by the social engineers: their idea is that by making sexual education normative, we can lower the rate of teen pregnancies and reduce the prevalence of STDs — it seems the majority of people fall in this camp. So, the legitimate question arises: what is the best policy of sex education to implement? It’s not too hard to objectify ‘best’: the best policy is the one which most lowers the rate of teen pregnancies and most reduces the prevalence of STDs.

Unfortunately, there are those like the author of this article who would shove Christian sexual ethics down the throat of wider society, in the name of beneficial social engineering. These folk are easily identifiable by their studied ignorance of the society-wide impact of an abstinence only policy, which can be measured empirically, in favor of best-case individual results.

That being said, the thesis of the article is that the abstinence only approach to teaching sex ed is more effective than approaches which additionally advocate condom use at preventing unwanted teen pregnancies. To support her claims, she appeals to this graph

which is pretty damning at first sight. If 17% of women who used condoms during intercourse got pregnant by age 20, it looks like condoms don’t work as well as advertised– I remember learning in sex ed that the failure rate for condoms is in the single digits. What’s going on?

Big surprise! It turns out that this graph is a misleading redrawing — note how ambiguously the title parses– of the original graph on page 19 of this CDC report (click for larger image if you need to)

which depicts the conception rate for women based on whether or not they used condoms the first time they had intercourse. Yep– just the first; after that, they could have been consistently unprotected, and they’d still be included in the data set.

According to the FDA, when condoms are used correctly during each act of intercourse, the expected failure rate is 3%; however, when condoms are incorrectly or inconsistently used, the failure rate jumps to 14%. Given these statistics, I think the townhall graphic is more a commentary on the fact that women who use condoms don’t use them consistently enough to reap their full contraceptive benefits.

Questions of whether the townhall graphic was deliberately mislabeled and misinterpreted or not aside, this study does not provide a basis for evaluating the efficacy of abstinence only vs. condom advocacy as approaches to sex ed. It is simply an illustration of the obvious: abstinence 100% of the time is a more effective contraceptive method than condom use some of the time. Really?!

If the author of this article wanted to convince me that abstinence only is better than condom advocacy, she’d do better to compare the percentage of pregnancies among teens who were educated in abstinence only programs versus the percentage of pregnancies among those who were educated in programs which offered condom use as an alternative. Until she, and others who support abstinence only programs can do that, I’ll stick to my belief that the best sex ed programs are those which deliver the same purely informational message as the NIH:

Sexual abstinence or sex with a single partner in a mutually monogamous, committed relationship remain the surest ways to prevent STDs, including HIV infection. Latex condoms should continue to be used consistently for other kinds of sexual partnerships.

I just finished chapter 2 in Fulton’s algebraic topology book, which covers winding numbers, homotopy, and vector fields. Along the way, I saw a theorem that reminded me an awful lot of the Cauchy-Goursat theorem in complex analysis: namely that if a function is sufficiently special, integrating it along two different contours which can be smoothly deformed into each other gives the same result. Specifically:

If and are smoothly homotopic closed paths in an open set , and is a closed 1-form in , then

Strictly speaking, the Cauchy-Goursat theorem says that the integral of an analytic function in a simple domain around a closed curve is zero:

Let be holomorphic in a simply connected domain and let be a piece-wise continuous closed path in , then

but some simple contour manipulations show that an easy equivalent statement is: if and are piece-wise continuous closed paths in , then

The connection between the two theorems was so striking that I did a little investigation, and found that homotopy theory provides a very concise and palatable proof of a (weaker?) version of the Cauchy-Goursat theorem:

Assume is analytic in an open set and are smoothly homotopic in with homotopy . Then .

To prove this, let represent the curve intermediate between determined by the blending parameter . We want to show that for all , is a constant. Note that . Since we’re assuming smoothness of , Liebniz’s rule gives

since . QED.

A powerful advantage of this version of the Cauchy-Goursat theorem is that the domain doesn’t need to be simply connected for the theorem to apply. It’s also clearer than the standard proofs I’ve seen.

Show that in a Banach space, three collinear points cannot all be the same distance from a fourth point. Or is this possible? I’m sure it isn’t in Hilbert spaces, but a proof eludes me in Banach spaces.

This came up as I’m trying to prove the convex projection/ best approximation theorem: given a closed convex set and a point , there exists a unique point in closest to .

As I was trying to recall anything about norms that might be useful in proving this, I realized that the law of cosines is a specific application of the polarization identity. In fact, if you write the law of cosines in the vector formulation (cf. the Wikipedia link) and replace the expression with , you get exactly the polarization identity. That was probably the motivation for the general Hilbert space result.

Another neat rephrasing of a more basic trig identity: .