Understanding branch cuts of expressions

Matthew England; Russell Bradford; James H. Davenport; David Wilson

doi:10.1007/978-3-642-39320-4_9

Understanding branch cuts of expressions

Matthew England, Russell Bradford, James H. Davenport, David Wilson

University of Bath

Research output: Chapter in Book/Report/Conference proceeding › Conference proceeding › peer-review

10 Citations (Scopus)

35 Downloads (Pure)

Abstract

We assume some standard choices for the branch cuts of a group of functions and consider the problem of then calculating the branch cuts of expressions involving those functions. Typical examples include the addition formulae for inverse trigonometric functions. Understanding these cuts is essential for working with the single-valued counterparts, the common approach to encoding multi-valued functions in computer algebra systems. While the defining choices are usually simple (typically portions of either the real or imaginary axes) the cuts induced by the expression may be surprisingly complicated. We have made explicit and implemented techniques for calculating the cuts in the computer algebra programme Maple. We discuss the issues raised, classifying the different cuts produced. The techniques have been gathered in the BranchCuts package, along with tools for visualising the cuts. The package is included in Maple 17 as part of the FunctionAdvisor tool.

Original language	English
Title of host publication	Intelligent Computer Mathematics - MKM, Calculemus, DML, and Systems and Projects 2013 - Held as Part of CICM 2013, Proceedings
Editors	Jacques Carette, David Aspinall, Christoph Lange, Petr Sojka, Wolfgang Windsteiger
Place of Publication	Heidelberg
Publisher	Springer
Pages	136-151
Number of pages	16
Volume	7961 LNAI
ISBN (Electronic)	9783642393204
ISBN (Print)	9783642393198
DOIs	https://doi.org/10.1007/978-3-642-39320-4_9
Publication status	Published - 2013
Externally published	Yes
Event	Conference on Intelligent Computer Mathematics, CICM 2013, Co-located with the MKM 2013, Calculemus 2013, DML 2013, and Systems and Projects 2013 - Bath, United Kingdom Duration: 8 Jul 2013 → 12 Jul 2013

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	7961 LNAI
ISSN (Print)	03029743
ISSN (Electronic)	16113349

Conference

Conference	Conference on Intelligent Computer Mathematics, CICM 2013, Co-located with the MKM 2013, Calculemus 2013, DML 2013, and Systems and Projects 2013
Country/Territory	United Kingdom
City	Bath
Period	8/07/13 → 12/07/13

Bibliographical note

The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-642-39320-4_9

Copyright © and Moral Rights are retained by the author(s) and/ or other copyright owners. A copy can be downloaded for personal non-commercial research or study, without prior permission or charge. This item cannot be reproduced or quoted extensively from without first obtaining permission in writing from the copyright holder(s). The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the copyright holders.

Keywords

branch cuts
simplification
symbolic computation

ASJC Scopus subject areas

Theoretical Computer Science
Computer Science(all)

Access to Document

10.1007/978-3-642-39320-4_9

Post-PrintAccepted author manuscript, 1.21 MBLicence: CC BY-NC

Cite this

England, M., Bradford, R., Davenport, J. H., & Wilson, D. (2013). Understanding branch cuts of expressions. In J. Carette, D. Aspinall, C. Lange, P. Sojka, & W. Windsteiger (Eds.), Intelligent Computer Mathematics - MKM, Calculemus, DML, and Systems and Projects 2013 - Held as Part of CICM 2013, Proceedings (Vol. 7961 LNAI, pp. 136-151). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 7961 LNAI). Springer. https://doi.org/10.1007/978-3-642-39320-4_9

Understanding branch cuts of expressions. / England, Matthew; Bradford, Russell; Davenport, James H. et al.

Intelligent Computer Mathematics - MKM, Calculemus, DML, and Systems and Projects 2013 - Held as Part of CICM 2013, Proceedings. ed. / Jacques Carette; David Aspinall; Christoph Lange; Petr Sojka; Wolfgang Windsteiger. Vol. 7961 LNAI Heidelberg : Springer, 2013. p. 136-151 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 7961 LNAI).

Research output: Chapter in Book/Report/Conference proceeding › Conference proceeding › peer-review

England, M, Bradford, R, Davenport, JH & Wilson, D 2013, Understanding branch cuts of expressions. in J Carette, D Aspinall, C Lange, P Sojka & W Windsteiger (eds), Intelligent Computer Mathematics - MKM, Calculemus, DML, and Systems and Projects 2013 - Held as Part of CICM 2013, Proceedings. vol. 7961 LNAI, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 7961 LNAI, Springer, Heidelberg, pp. 136-151, Conference on Intelligent Computer Mathematics, CICM 2013, Co-located with the MKM 2013, Calculemus 2013, DML 2013, and Systems and Projects 2013, Bath, United Kingdom, 8/07/13. https://doi.org/10.1007/978-3-642-39320-4_9

England M, Bradford R, Davenport JH, Wilson D. Understanding branch cuts of expressions. In Carette J, Aspinall D, Lange C, Sojka P, Windsteiger W, editors, Intelligent Computer Mathematics - MKM, Calculemus, DML, and Systems and Projects 2013 - Held as Part of CICM 2013, Proceedings. Vol. 7961 LNAI. Heidelberg: Springer. 2013. p. 136-151. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/978-3-642-39320-4_9

England, Matthew ; Bradford, Russell ; Davenport, James H. et al. / Understanding branch cuts of expressions. Intelligent Computer Mathematics - MKM, Calculemus, DML, and Systems and Projects 2013 - Held as Part of CICM 2013, Proceedings. editor / Jacques Carette ; David Aspinall ; Christoph Lange ; Petr Sojka ; Wolfgang Windsteiger. Vol. 7961 LNAI Heidelberg : Springer, 2013. pp. 136-151 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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abstract = "We assume some standard choices for the branch cuts of a group of functions and consider the problem of then calculating the branch cuts of expressions involving those functions. Typical examples include the addition formulae for inverse trigonometric functions. Understanding these cuts is essential for working with the single-valued counterparts, the common approach to encoding multi-valued functions in computer algebra systems. While the defining choices are usually simple (typically portions of either the real or imaginary axes) the cuts induced by the expression may be surprisingly complicated. We have made explicit and implemented techniques for calculating the cuts in the computer algebra programme Maple. We discuss the issues raised, classifying the different cuts produced. The techniques have been gathered in the BranchCuts package, along with tools for visualising the cuts. The package is included in Maple 17 as part of the FunctionAdvisor tool.",

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note = "The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-642-39320-4_9 Copyright {\textcopyright} and Moral Rights are retained by the author(s) and/ or other copyright owners. A copy can be downloaded for personal non-commercial research or study, without prior permission or charge. This item cannot be reproduced or quoted extensively from without first obtaining permission in writing from the copyright holder(s). The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the copyright holders. ; Conference on Intelligent Computer Mathematics, CICM 2013, Co-located with the MKM 2013, Calculemus 2013, DML 2013, and Systems and Projects 2013 ; Conference date: 08-07-2013 Through 12-07-2013",

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Understanding branch cuts of expressions

Abstract

Publication series

Conference

Bibliographical note

Keywords

ASJC Scopus subject areas

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Cite this