Inevitable Dottie Number. Iterals of cosine and sine

• Published in 2012
• Added on 2026-10-14

The unique real root of cos(x) = x, recently referred to as the Dottie number, is expressed as an iteral of cosine. Using the derivatives of iterals, it is shown why this number is achieved starting from any real number, when the iterates of cosine successfully approach infinity, and how this affects the Maclaurin series of the iterals. Properties of the iterals of cosine and sine and their derivatives are considered. A C++ template for iteral is applied for computation of Julia sets.

Links

• https://arxiv.org/abs/1212.1027v1
• https://arxiv.org/pdf/1212.1027v1

Other information

BibTeX entry

@article{InevitableDottieNumberIteralsofcosineandsine,
	key = {InevitableDottieNumberIteralsofcosineandsine},
	type = {article},
	title = {Inevitable Dottie Number. Iterals of cosine and sine},
	author = {Valerii Salov},
	abstract = {The unique real root of cos(x) = x, recently referred to as the Dottie number, is expressed as an iteral of cosine. Using the derivatives of iterals, it is shown why this number is achieved starting from any real number, when the iterates of cosine successfully approach infinity, and how this affects the Maclaurin series of the iterals. Properties of the iterals of cosine and sine and their derivatives are considered. A C++ template for iteral is applied for computation of Julia sets.},
	comment = {},
	date_added = {2026-10-14},
	date_published = {2012-10-14},
	urls = {https://arxiv.org/abs/1212.1027v1,https://arxiv.org/pdf/1212.1027v1},
	collections = {easily-explained,fun-maths-facts},
	url = {https://arxiv.org/abs/1212.1027v1 https://arxiv.org/pdf/1212.1027v1},
	year = 2012,
	urldate = {2026-10-14},
	archivePrefix = {arXiv},
	eprint = {1212.1027},
	primaryClass = {math.HO}
}