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Schwarzian Derivative: Simple Definition, Examples

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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 [1].

Although it’s called a “derivative”, a Shwarzian isn’t a generalization of the ordinary derivative but rather a differential operator analogous to the ordinary derivative.

The name is due to Cayley [2], who named the derivative after German mathematician Hermann Schwarz.

Definition of the Schwarzian Derivative

Formally, the Shwarzian is defined as [3]:
shwarzian derivative

Where:

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

Möbius transformations (also called fractional linear transformations) are functions of the form:
mobius transformations

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.

References

[1] 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
[2] Cayley, A. (1880). On the Schwarzian derivative and the polyhedral functions. Trans. Camb. Phil. Soc., 13.
[3] Ovsienko, V. & Tabachnikov, S. What is the Schwarzian Derivative? Retrieved July 7, 2021 from: https://www.ams.org/notices/200901/tx090100034p.pdf

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