A chronology of continued square roots

“The theory of continued radicals is far from new, and M[onsieur] A. Bouché is not the first to have studied it.”

— S. Realis, 1877 *

I’ve compiled a bibliography of published work concerning

  • continued square roots, also known as continued roots, continued radicals, iterated radicals, infinite radicals, infinite(ly) nested radicals, Kettenwurzeln, Quadratwurzeloperationen, radicales infinitas, catene di radicali quadratici, racines continues, radicaux continus, and radicaux superposés;
  • continued powers, including continued reciprocal powers, continued reciprocal roots, reziproker Kettenpotenzen, and reziproker Kettenwurzeln;
  • continued cotangents; and
  • continued function compositions, including f-expansions, chain operations, Kettenoperationen, and continued operations.

The document, named “A chronology of continued square roots and other continued compositions, through the year 2016,” is available at


Version 2 of this arXiv document was uploaded on 15 June 2018.

An earlier version of this bibliography, posted beginning in 2008 on a personal website and at academia.edu, was called “A chronology of continued radicals and other selected chain compositions.”

I apologize for the disappearance of the link


that appeared as a reference in my paper “Continued reciprocal roots,” Ramanujan J. The content contained in this defunct reference is expanded upon in the arXiv link above.

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Updated 5 Sept 2021