What is the Schwarzian derivative?
The Schwarzian derivative or just “Schwarzian” is, informally speaking, curvature. In complex analysis, it measures how well a Möbius transformation approximates a function. It also gives some other useful information about behavior of functions, especially at critical points .
The name is due to Cayley , who named the derivative after German mathematician Hermann Schwarz.
Definition of the Schwarzian Derivative
Formally, the Shwarzian is defined as :
- f(x) = a real-valued or complex-valued function of one variable,
- f′(x) = first derivative,
- f′′(x) = second derivative,
- f′′′(x) = third derivative.
Critical Points and the Shwarzian
The Schwarzian gives particularly useful information when it is negative. Let’s say x0 is an attracting periodic point of f. A negative Schwarzian tells us one of two things:
- Either the immediate basin of attraction of x0 extends to +∞ or −∞, or
- A critical point of f has an orbit attracted to the orbit of x0.
Möbius transformations (also called fractional linear transformations) are functions of the form:
These transformations roughly equate to the constant in an ordinary derivative. Constant functions have ordinary derivatives of zero; In the same way, if a function’s Schwarzian derivative is zero, then that function is a Möbius transformation. Adding a constant to such a function doesn’t change its Schwarzian derivative, nor does multiplying by a constant. With ordinary derivatives, you can pull a constant outside of the function; When you pull a Möbius transformation outside, it just disappears.
 McKinney, W. (2005). The Schwarzian Derivative & the Critical Orbit. Retrieved July 7, 2021 from: https://ocw.mit.edu/courses/mathematics/18-091-mathematical-exposition-spring-2005/lecture-notes/lecture09.pdf
 Cayley, A. (1880). On the Schwarzian derivative and the polyhedral functions. Trans. Camb. Phil. Soc., 13.
 Ovsienko, V. & Tabachnikov, S. What is the Schwarzian Derivative? Retrieved July 7, 2021 from: https://www.ams.org/notices/200901/tx090100034p.pdf
Stephanie Glen. "Schwarzian Derivative: Simple Definition, Examples" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/schwarzian-derivative-simple-definition-examples/
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