The Witch of Agnesi $y = \dfrac{1}{1+x^2}$

$y = \dfrac{1}{1+x^2}$ is a rational function known as the Witch of Agnesi¹, and its graph is a bell shape closely resembling the Gaussian. Because its denominator is a polynomial rather than an exponential, its tails fall off more gently than the Gaussian's.

Domain and range

• The domain is all real numbers
• The range is $0 < y \leq 1$
• Even function
• The maximum $1$ is attained at the origin

The denominator $1 + x^2$ is always at least $1$ and never zero, so the function is defined everywhere.

Monotonicity and extrema

The derivative is $f'(x) = \dfrac{-2x}{(1+x^2)^2}$. It is positive for $x < 0$ and negative for $x > 0$, so the curve reaches its maximum at the peak $(0, 1)$ and decreases on either side.

Asymptote and decay

As $x \to \pm\infty$ the denominator grows without bound and $y \to 0$, so the $x$-axis is a horizontal asymptote.

The decay is only of order $\dfrac{1}{x^2}$, so the tails stay far heavier than the Gaussian's.

Inflection points

The second derivative $f''(x) = \dfrac{6x^2 - 2}{(1+x^2)^3}$ changes sign at $x = \pm \dfrac{1}{\sqrt{3}} \approx \pm 0.577$, where $y = \dfrac{3}{4}$. These shoulders mark the change from concave down to concave up.

Relation to other functions

This function is exactly the derivative of the arctangent.

Its total integral is $\int_{-\infty}^{\infty}\dfrac{dx}{1+x^2} = \pi$, and dividing by $\pi$ gives $\dfrac{1}{\pi(1+x^2)}$, the probability density of the Cauchy distribution².

History and applications

The curve is named after the eighteenth-century Italian mathematician Maria Gaetana Agnesi. When it appeared in her $1748$ textbook, the Italian word for a turning curve was confused with the word for a she-devil, leading to the famous English mistranslation "witch". Today it arises as the Cauchy distribution in probability and the Lorentzian function describing resonance.

1. Witch of Agnesi, Wikipedia
2. Cauchy distribution, Wikipedia