Thread by @TamasGorbe

Here is how you draw the Witch of Agnesi. It's a nice bell-shaped curve that is used as the probability density for the Cauchy distribution. Surprisingly, the region between the curve and its asymptote has no balance point (centroid) [Wiki bit.ly/2TjqYGA] #50FamousCurves

Here is the reason it doesn't have a balance point: The x-coordinate of the centroid of a region between the x axis and the graph of a function y=y(x), A<x<B is x̅ = ∫ x y(x)dx / ∫ y(x)dx, where A and B are the bounds of the definite integrals.

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In the case of the Witch of Agnesi, we have y(x)=a³/(x²+a²) with some fixed a>0 and A=-∞, B=∞. The indefinite integrals are easy to calculate:
∫ y(x)dx = ∫ a³/(x²+a²)dx = a² arctan(x/a)+C
and
∫ x y(x)dx = ∫ x a³/(x²+a²)dx = a³ ln(x²+a²)/2+C'.

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Next we calculate the definite integrals. These are improper integrals because their bounds are at ±∞. The first one is well-defined due to arctan(x)→±π/2 as x→±∞, so the area of the region is finite, namely a²π. It's 4× the area of the circle used in the construction.

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The problem is the second improper integral. It's not well-defined, because its value depends on the way we take the limits of ln(x²+a²) as x→±∞. For example, taking ln(L²+a²)-ln((-L)²+a²) and L→∞ gives 0, but if we take ln((2L)²+a²)-ln((-L)²+a²) and L→∞ we get 2ln(2).

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Fun fact: The name "witch" derives from a mistranslation of the term "averisera" meaning versed sine curve as "avversiera" meaning witch or wife of the devil. #50FamousCurves

Fun fact: X-ray emission lines can be represented very accurately by the Witch of Agnesi. #50FamousCurves [Source: doi.org/10.1103/PhysRe…]

Fun fact: The Poisson kernel for the upper half-plane y>0 is the rescaled Witch of Agnesi Pʸ(x)=y/[π(x²+y²)]. This means that heating the edge of the half-plane according to a function f(x) leads to the heat distribution f * Pʸ on the half-plane (*=convolution) . #50FamousCurves

In the animation above f(x)=1 for |x|≤1/2 and f(x)=0 for |x|>1/2. You can see how the convolution with Pʸ gives rise to a smooth heat distribution on the upper half-plane. #50FamousCurves

Fun fact: If you wrap the graph of the tangent function around a cylinder and look at it from the side, you'll see the Witch of Agnesi. #50FamousCurves