As we get closer to Halloween, let's take a look at one of the few functions that might cause a stir in Salem. I give you the Witch of Agnesi, for several values of its free parameter a:
The Witch is a geometric construction involving triangles that's easier to show than describe:
(Both images are from the Wikipedia article)
Geometric construction though it is, you can also write it in closed form as a standard old function of x:
Why did this seemingly mundane function merit such a creepy name? It was originally named by the Italian mathematician Maria Gaetana Agnesi, who called it la versiera, apparently a nautical term meaning "a rope which turns a sail". This fits pretty well with the geometric picture of how the curve is drawn. Somehow the English translator managed to read la versiera as l'avversiera - a she-devil. To this day we're still calling this curve "The Witch of Agnesi".
The curve happens to represent a probability density function useful in physics and elsewhere. The particular distribution is the Cauchy distribution. It's a little screwy since neither its mean nor its variance are well-defined. It also happens to describe a driven oscillator near resonance.
It's a nice function to have around. I don't expect physics will have to summon the Spanish Inquisition on it!
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I thought the variance is well defined but that it's infinity. It crops up quite a bit in stats textbooks, as a counter-example to limit theorems, for example the Law of Large Numbers.
It's kind of a matter of preference. Formally infinity isn't a number, so saying an expression "equals infinity" is a shorthand for a specific type of limit behavior. In physics nobody cares about the distinction, but mathematicians might look askance a about saying the variance was infinite without further formal description of just what that means in this context.
No, neither the mean or the variance exist, because the required integrals diverge (to infinity).
As a counter example to limit theorems, the Cauchy distribution provides this: if you have the "standard cauchy" (peak at zero, scale parameter =1), the mean of a random sample of size n has the same distribution - a glaring difference to what we are used to seeing.
Agnesi (1718 - 1799) is of course a famous example of an early female mathematician who was so successful that she was offered a chair at the University of Bologna in 1750. Bologna seems in the 18th century to have been particularly progressive in terms of female appointments as it appointed one of Agnesi's contemporaries Laura Bassi (1711 - 1778) to the chair of philosophy in 1734 where she was a key figure in the introduction of Newtonian physics into Italy.
Well, nobody expects the Spanish Inquisition!
that animation is lovely.
thanks, merril
http://commons.wikimedia.org/wiki/User:Merrill
wikipedia is fantastic. i wonder if kids will be able to learn more, because information is easier to obtain, or just get intellectually lazy, becuase no effort is required to find the information.
Also, if you leave the circle in the plot at the end it looks (vaguely) like a face with hair flowing out both sides! Now we just need a pointy hat to go on top. Perhaps y = c - x ?