cis (mathematics)

$cis$ is a mathematical notation defined by $cis x = cos x + i sin x$ ,^{[nb 1]} where $cos$ is the cosine function, $i$ is the imaginary unit and $sin$ is the sine function. $x$ is the argument of the complex number (angle between line to point and x-axis in polar form). The notation is less commonly used in mathematics than Euler's formula, $e ix$ , which offers an even shorter notation for $cos x + i sin x$ , but cis(x) is widely used as a name for this function in software libraries.

Overview

The $cis$ notation is a shorthand for the combination of functions on the right-hand side of Euler's formula:

e^{ix}=\cos x+i\sin x,

where $i 2 = -1$ . So,

\operatorname {cis} x=\cos x+i\sin x,

^[1]^[2]^[3]^[4]

i.e. " $cis$ " is an acronym for " $Cos i Sin$ ".

It connects trigonometric functions with exponential functions in the complex plane via Euler's formula. While the domain of definition is usually $x\in \mathbb {R}$ , complex values $z\in \mathbb {C}$ are possible as well:

\operatorname {cis} z=\cos z+i\sin z,

so the $cis$ function can be used to extend Euler's formula to a more general complex version.^[5]

The function is mostly used as a convenient shorthand notation to simplify some expressions,^[6]^[7]^[8] for example in conjunction with Fourier and Hartley transforms,^[9]^[10]^[11] or when exponential functions shouldn't be used for some reason in math education.

In information technology, the function sees dedicated support in various high-performance math libraries (such as Intel's Math Kernel Library (MKL)^[12] or MathCW^[13]), available for many compilers and programming languages (including C, C++,^[14] Common Lisp,^[15]^[16] D,^[17] Haskell,^[18] Julia,^[19] and Rust^[20]). Depending on the platform, the fused operation is about twice as fast as calling the sine and cosine functions individually.^[17]^[3]

Mathematical identities

Derivative

{\frac {\mathrm {d} }{\mathrm {d} z}}\operatorname {cis} z=i\operatorname {cis} z=ie^{iz}

^[1]^[21]

Integral

\int \operatorname {cis} z\,\mathrm {d} z=-i\operatorname {cis} z=-ie^{iz}

^[1]

Other properties

These follow directly from Euler's formula.

\cos(x)={\frac {\operatorname {cis} (x)+\operatorname {cis} (-x)}{2}}={\frac {e^{ix}+e^{-ix}}{2}}

\sin(x)={\frac {\operatorname {cis} (x)-\operatorname {cis} (-x)}{2i}}={\frac {e^{ix}-e^{-ix}}{2i}}

\operatorname {cis} (x+y)=\operatorname {cis} x\,\operatorname {cis} y

^[22]

\operatorname {cis} (x-y)={\operatorname {cis} x \over \operatorname {cis} y}

The identities above hold if $x$ and $y$ are any complex numbers. If $x$ and $y$ are real, then

|\operatorname {cis} x-\operatorname {cis} y|\leq |x-y|.

^[22]

History

The $cis$ notation was first coined by William Edwin Hamilton in Elements of Quaternions (1866)^[23]^[24] and subsequently used by Irving Stringham (who also called it "sector of $x$ ") in works such as Uniplanar Algebra (1893),^[25]^[26] James Harkness and Frank Morley in their Introduction to the Theory of Analytic Functions (1898),^[26]^[27] or by George Ashley Campbell (who also referred to it as "cisoidal oscillation") in his works on transmission lines (1901) and Fourier integrals (1928).^[28]^[29]^[30]

In 1942, inspired by the $cis$ notation, Ralph V. L. Hartley introduced the $cas$ (for cosine-and-sine) function for the real-valued Hartley kernel, a meanwhile established shortcut in conjunction with Hartley transforms:^[31]^[32]

\operatorname {cas} x=\cos x+\sin x.

Motivation

The $cis$ notation is sometimes used to emphasize one method of viewing and dealing with a problem over another.^[33] The mathematics of trigonometry and exponentials are related but not exactly the same; exponential notation emphasizes the whole, whereas $cis x$ and $cos x + i sin x$ notations emphasize the parts. This can be rhetorically useful to mathematicians and engineers when discussing this function, and further serve as a mnemonic (for $cos + i sin$ ).^[30]

The $cis$ notation is convenient for math students whose knowledge of trigonometry and complex numbers permit this notation, but whose conceptual understanding does not yet permit the notation $e ix$ . The usual proof that $cis x = e ix$ requires calculus, which the student may not have studied before encountering the expression $cos x + i sin x$ .

This notation was more common when typewriters were used to convey mathematical expressions.^{[citation needed]}

Notes

^ Here, $i$ refers to the imaginary unit in mathematics. Since $i$ is commonly used to denote electric current in electrical engineering and control systems engineering, the imaginary unit is alternatively denoted by $j$ instead of $i$ in these contexts. Regardless of context, this does not affect the established name of the function as $cis$ .

References

^ ^a ^b ^c Weisstein, Eric Wolfgang (2015) [2000]. "Cis". MathWorld. Wolfram Research, Inc. Archived from the original on 2016-01-27. Retrieved 2016-01-09.
^ Simmons, Bruce (2014-07-28) [2004]. "Cis". Mathwords: Terms and Formulas from Algebra I to Calculus. Oregon City, Oregon, USA: Clackamas Community College, Mathematics Department. Archived from the original on 2023-07-16. Retrieved 2016-01-15.
^ ^a ^b "Rationale for International Standard - Programming Languages - C" (PDF). 5.10. April 2003. pp. 114, 117, 183, 186–187. Archived (PDF) from the original on 2016-06-06. Retrieved 2010-10-17.
^ Amann, Herbert [at Wikidata]; Escher, Joachim [in German] (2006). Analysis I. Grundstudium Mathematik (in German) (3 ed.). Basel, Switzerland: Birkhäuser Verlag. pp. 292, 298. ISBN 978-3-76437755-7. ISBN 3-76437755-0. (445 pages)
^ Moskowitz, Martin A. (2002). "Chapter 1. First Concepts". Written at City University of New York Graduate Center, New York, USA. A Course in Complex Analysis in One Variable. Singapore: World Scientific Publishing Co. Pte. Ltd. p. 7. ISBN 981-02-4780-X. (ix+149 pages)
^ Swokowski, Earl William [at Wikidata]; Cole, Jeffery (2011). Precalculus: Functions and Graphs. Precalculus Series (12 ed.). Cengage Learning. ISBN 978-0-84006857-6. Retrieved 2016-01-18.
^ Reis, Clive (2011). Abstract Algebra: An Introduction to Groups, Rings and Fields (1 ed.). World Scientific Publishing Co. Pte. Ltd. pp. 434–438. ISBN 978-9-81433564-5.
^ Weitz, Edmund [in German] (2016). "The fundamental theorem of algebra - a visual proof". Hamburg, Germany: Hamburg University of Applied Sciences (HAW), Department Medientechnik. Archived from the original on 2019-08-03. Retrieved 2019-08-03.
^ L.-Rundblad, Ekaterina; Maidan, Alexei; Novak, Peter; Labunets, Valeriy (2004). "Fast Color Wavelet-Haar-Hartley-Prometheus Transforms for Image Processing". Written at Prometheus Inc., Newport, USA. In Byrnes, Jim (ed.). Computational Noncommutative Algebra and Applications (PDF). NATO Science Series II: Mathematics, Physics and Chemistry (NAII). Vol. 136. Dordrecht, Netherlands: Springer Science + Business Media, Inc. pp. 401–411. doi:10.1007/1-4020-2307-3. ISBN 978-1-4020-1982-1. ISSN 1568-2609. Archived (PDF) from the original on 2017-10-28. Retrieved 2017-10-28.
^ Kammler, David W. (2008-01-17). A First Course in Fourier Analysis (2 ed.). Cambridge University Press. ISBN 978-1-13946903-6. Retrieved 2017-10-28.
^ Lorenzo, Carl F.; Hartley, Tom T. (2016-11-14). The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science. John Wiley & Sons. ISBN 978-1-11913942-3. Retrieved 2017-10-28.
^ "v?CIS". Developer Reference for Intel Math Kernel Library (Intel MKL) 2017 - C. MKL documentation; IDZ Documentation Library. Intel Corporation. 2016-09-06. p. 1799. 671504. Retrieved 2016-01-15.
^ Beebe, Nelson H. F. (2017-08-22). "Chapter 15.2. Complex absolute value". The Mathematical-Function Computation Handbook - Programming Using the MathCW Portable Software Library (1 ed.). Salt Lake City, Utah, USA: Springer International Publishing AG. p. 443. doi:10.1007/978-3-319-64110-2. ISBN 978-3-319-64109-6. LCCN 2017947446. S2CID 30244721.
^ "Intel C++ Compiler Reference" (PDF). Intel Corporation. 2007 [1996]. pp. 34, 59–60. 307777-004US. Archived (PDF) from the original on 2023-07-16. Retrieved 2016-01-15.
^ "CIS". Common Lisp Hyperspec. The Harlequin Group Limited. 1996. Archived from the original on 2023-07-16. Retrieved 2016-01-15.
^ "CIS". LispWorks, Ltd. 2005 [1996]. Archived from the original on 2023-07-16. Retrieved 2016-01-15.
^ ^a ^b "std.math: expi". D programming language. Digital Mars. 2016-01-11 [2000]. Archived from the original on 2023-07-16. Retrieved 2016-01-14.
^ "CIS". Haskell reference. ZVON. Archived from the original on 2023-07-16. Retrieved 2016-01-15.
^ "Mathematics; Mathematical Operators". The Julia Language. Archived from the original on 2020-08-19. Retrieved 2019-12-05.
^ "Struct num_complex::Complex". Archived from the original on 2023-07-16. Retrieved 2022-08-05.
^ Fuchs, Martin (2011). "Chapter 11: Differenzierbarkeit von Funktionen". Analysis I (PDF) (in German) (WS 2011/2012 ed.). Fachrichtung 6.1 Mathematik, Universität des Saarlandes, Germany´. pp. 3, 13. Archived (PDF) from the original on 2023-07-16. Retrieved 2016-01-15.
^ ^a ^b Fuchs, Martin (2011). "Chapter 8.IV: Spezielle Funktionen – Die trigonometrischen Funktionen". Analysis I (PDF) (in German) (WS 2011/2012 ed.). Fachrichtung 6.1 Mathematik, Universität des Saarlandes, Germany. pp. 16–20. Archived (PDF) from the original on 2023-07-16. Retrieved 2016-01-15.
^ Hamilton, William Rowan (1866-01-01). "Book II, Chapter II. Fractional powers, General roots of unity". Written at Dublin, Irland. In Hamilton, William Edwin (ed.). Elements of Quaternions (1 ed.). London, UK: Longmans, Green & Co., University Press, Michael Henry Gill. pp. 250–257, 260, 262–263. Retrieved 2016-01-17. pp. 250, 252: [...] cos [...] + i sin [...] we shall occasionally abridge to the following: [...] cis [...]. As to the marks [...], they are to be considered as chiefly available for the present exposition of the system, and as not often wanted, nor employed, in the subsequent practise thereof; and the same remark applies to the recent abrigdement cis, for cos + i sin [...] ([1], [2][3]) (NB. This work was published posthumously, Hamilton died in 1865.)
^ Hamilton, William Rowan (1899) [1866-01-01]. Hamilton, William Edwin; Joly, Charles Jasper (eds.). Elements of Quaternions. Vol. I (2 ed.). London, UK: Longmans, Green & Co. p. 262. Retrieved 2019-08-03. p. 262: [...] recent abridgment cis for cos + i sin [...] (NB. This edition was reprinted by Chelsea Publishing Company in 1969.)
^ Stringham, Irving (1893-07-01) [1891]. Uniplanar Algebra, being part 1 of a propædeutic to the higher mathematical analysis. Vol. 1. C. A. Mordock & Co. (printer) (1 ed.). San Francisco, California, USA: The Berkeley Press. pp. 71–75, 77, 79–80, 82, 84–86, 89, 91–92, 94–95, 100–102, 116, 123, 128–129, 134–135. Retrieved 2016-01-18. p. 71: As an abbreviation for cos θ + i sin θ it is convenient to use cis θ, which may be read: sector of θ.
^ ^a ^b Cajori, Florian (1952) [March 1929]. A History of Mathematical Notations. Vol. 2 (3rd corrected printing of 1929 issue, 2nd ed.). Chicago, Illinois, USA: Open court publishing company. p. 133. ISBN 978-1-60206-714-1. Retrieved 2016-01-18. p. 133: Stringham denoted cos β + i sin β by "cis β", a notation also used by Harkness and Morley. (NB. ISBN and link for reprint of 2nd edition by Cosimo, Inc., New York, US, 2013.)
^ Harkness, James; Morley, Frank (1898). Introduction to the Theory of Analytic Functions (1 ed.). London, UK: Macmillan and Company. pp. 18, 22, 48, 52, 170. ISBN 978-1-16407019-1. Retrieved 2016-01-18. (NB. ISBN for reprint by Kessinger Publishing, 2010.)
^ Campbell, George Ashley (1903) [1901-06-07]. "Chapter XXX. On loaded lines in telephonic transmission" (PDF). The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. Series 6. 5 (27). Taylor & Francis: 313–330. doi:10.1080/14786440309462928. Archived (PDF) from the original on 2023-07-16. Retrieved 2023-07-16. (2+18 pages)
^ Campbell, George Ashley (April 1911). "Cisoidal oscillations" (PDF). Proceedings of the American Institute of Electrical Engineers. XXX (1–6). American Institute of Electrical Engineers: 789–824. doi:10.1109/PAIEE.1911.6659711. S2CID 51647814. Retrieved 2023-06-24. (37 pages)
^ ^a ^b Campbell, George Ashley (1928-10-01) [1927-09-13]. "The Practical Application of the Fourier Integral" (PDF). The Bell System Technical Journal. 7 (4). American Telephone and Telegraph Company: 639–707 [641]. doi:10.1002/j.1538-7305.1928.tb00347.x. S2CID 53552671. Retrieved 2023-06-24. p. 641: It has been recognized, almost from the start, however, that the form which best combines mathematical simplicity and complete generality makes use of the exponential oscillating function e^i2πft. More recently the overwhelming advantage of using this oscillating function in the discussion of sinusoidal oscillatory systems has been generally recognized. It is, therefore, plain that this oscillating function should be adopted as the basic oscillation for both of the proposed tables. A name for this oscillation, associating it with sines and cosines, rather than with the real exponential function, seems desirable. The abbreviation cis x for (cos x + i sin x) suggests that we name this function a cis or a cisoidal oscillation. (69 pages)
^ Hartley, Ralph V. L. (March 1942). "A More Symmetrical Fourier Analysis Applied to Transmission Problems". Proceedings of the IRE. 30 (3). Institute of Radio Engineers: 144–150. doi:10.1109/JRPROC.1942.234333. S2CID 51644127. Archived from the original on 2019-04-05. Retrieved 2023-07-16.
^ Bracewell, Ronald N. (June 1999) [1985, 1978, 1965]. The Fourier Transform and Its Applications (3 ed.). McGraw-Hill. ISBN 978-0-07303938-1.
^ Diehl, Christina; Leupp, Marcel (January 2010). Komplexe Zahlen: Ein Leitprogramm in Mathematik (PDF) (in German). Basel & Herisau, Switzerland: Eidgenössische Technische Hochschule Zürich (ETH). p. 41. Archived (PDF) from the original on 2017-08-27. p. 41: [...] Bitte vergessen Sie aber nicht, dass e^iφ für uns bisher nur eine Schreibweise für den Einheitszeiger mit Winkel φ ist. In anderen Büchern wird dafür oft der Ausdruck cis(φ) anstelle von e^iφ verwendet. [...] (109 pages)

[NB_ij-1] Here, $i$ refers to the imaginary unit in mathematics. Since $i$ is commonly used to denote electric current in electrical engineering and control systems engineering, the imaginary unit is alternatively denoted by $j$ instead of $i$ in these contexts. Regardless of context, this does not affect the established name of the function as $cis$ .

[Weisstein_cis-2] Weisstein, Eric Wolfgang (2015) [2000]. "Cis". MathWorld. Wolfram Research, Inc. Archived from the original on 2016-01-27. Retrieved 2016-01-09.

[Simmons_2014_1-3] Simmons, Bruce (2014-07-28) [2004]. "Cis". Mathwords: Terms and Formulas from Algebra I to Calculus. Oregon City, Oregon, USA: Clackamas Community College, Mathematics Department. Archived from the original on 2023-07-16. Retrieved 2016-01-15.

[Rationale_2003_C-4] "Rationale for International Standard - Programming Languages - C" (PDF). 5.10. April 2003. pp. 114, 117, 183, 186–187. Archived (PDF) from the original on 2016-06-06. Retrieved 2010-10-17.

[Amann-Escher_2006-5] Amann, Herbert [at Wikidata]; Escher, Joachim [in German] (2006). Analysis I. Grundstudium Mathematik (in German) (3 ed.). Basel, Switzerland: Birkhäuser Verlag. pp. 292, 298. ISBN 978-3-76437755-7. ISBN 3-76437755-0. (445 pages)

[Moskowitz_2002-6] Moskowitz, Martin A. (2002). "Chapter 1. First Concepts". Written at City University of New York Graduate Center, New York, USA. A Course in Complex Analysis in One Variable. Singapore: World Scientific Publishing Co. Pte. Ltd. p. 7. ISBN 981-02-4780-X. (ix+149 pages)

[Swokowski_2011-7] Swokowski, Earl William [at Wikidata]; Cole, Jeffery (2011). Precalculus: Functions and Graphs. Precalculus Series (12 ed.). Cengage Learning. ISBN 978-0-84006857-6. Retrieved 2016-01-18.

[Reis_2011-8] Reis, Clive (2011). Abstract Algebra: An Introduction to Groups, Rings and Fields (1 ed.). World Scientific Publishing Co. Pte. Ltd. pp. 434–438. ISBN 978-9-81433564-5.

[Weitz_2016-9] Weitz, Edmund [in German] (2016). "The fundamental theorem of algebra - a visual proof". Hamburg, Germany: Hamburg University of Applied Sciences (HAW), Department Medientechnik. Archived from the original on 2019-08-03. Retrieved 2019-08-03.

[Byrnes_2004-10] L.-Rundblad, Ekaterina; Maidan, Alexei; Novak, Peter; Labunets, Valeriy (2004). "Fast Color Wavelet-Haar-Hartley-Prometheus Transforms for Image Processing". Written at Prometheus Inc., Newport, USA. In Byrnes, Jim (ed.). Computational Noncommutative Algebra and Applications (PDF). NATO Science Series II: Mathematics, Physics and Chemistry (NAII). Vol. 136. Dordrecht, Netherlands: Springer Science + Business Media, Inc. pp. 401–411. doi:10.1007/1-4020-2307-3. ISBN 978-1-4020-1982-1. ISSN 1568-2609. Archived (PDF) from the original on 2017-10-28. Retrieved 2017-10-28.

[Kammler_2008-11] Kammler, David W. (2008-01-17). A First Course in Fourier Analysis (2 ed.). Cambridge University Press. ISBN 978-1-13946903-6. Retrieved 2017-10-28.

[Lorenzo-Hartley_2016-12] Lorenzo, Carl F.; Hartley, Tom T. (2016-11-14). The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science. John Wiley & Sons. ISBN 978-1-11913942-3. Retrieved 2017-10-28.

[Intel_MKL-13] "v?CIS". Developer Reference for Intel Math Kernel Library (Intel MKL) 2017 - C. MKL documentation; IDZ Documentation Library. Intel Corporation. 2016-09-06. p. 1799. 671504. Retrieved 2016-01-15.

[Beebe_2017-14] Beebe, Nelson H. F. (2017-08-22). "Chapter 15.2. Complex absolute value". The Mathematical-Function Computation Handbook - Programming Using the MathCW Portable Software Library (1 ed.). Salt Lake City, Utah, USA: Springer International Publishing AG. p. 443. doi:10.1007/978-3-319-64110-2. ISBN 978-3-319-64109-6. LCCN 2017947446. S2CID 30244721.

[Intel_2007_C++-15] "Intel C++ Compiler Reference" (PDF). Intel Corporation. 2007 [1996]. pp. 34, 59–60. 307777-004US. Archived (PDF) from the original on 2023-07-16. Retrieved 2016-01-15.

[Lisp-16] "CIS". Common Lisp Hyperspec. The Harlequin Group Limited. 1996. Archived from the original on 2023-07-16. Retrieved 2016-01-15.

[Lisp_2005-17] "CIS". LispWorks, Ltd. 2005 [1996]. Archived from the original on 2023-07-16. Retrieved 2016-01-15.

[D_2011-18] "std.math: expi". D programming language. Digital Mars. 2016-01-11 [2000]. Archived from the original on 2023-07-16. Retrieved 2016-01-14.

[Haskell-19] "CIS". Haskell reference. ZVON. Archived from the original on 2023-07-16. Retrieved 2016-01-15.

[Julia-20] "Mathematics; Mathematical Operators". The Julia Language. Archived from the original on 2020-08-19. Retrieved 2019-12-05.

[Rust-21] "Struct num_complex::Complex". Archived from the original on 2023-07-16. Retrieved 2022-08-05.

[Fuchs_2011_2-22] Fuchs, Martin (2011). "Chapter 11: Differenzierbarkeit von Funktionen". Analysis I (PDF) (in German) (WS 2011/2012 ed.). Fachrichtung 6.1 Mathematik, Universität des Saarlandes, Germany´. pp. 3, 13. Archived (PDF) from the original on 2023-07-16. Retrieved 2016-01-15.

[Fuchs_2011_1-23] Fuchs, Martin (2011). "Chapter 8.IV: Spezielle Funktionen – Die trigonometrischen Funktionen". Analysis I (PDF) (in German) (WS 2011/2012 ed.). Fachrichtung 6.1 Mathematik, Universität des Saarlandes, Germany. pp. 16–20. Archived (PDF) from the original on 2023-07-16. Retrieved 2016-01-15.

[Hamilton_1866-24] Hamilton, William Rowan (1866-01-01). "Book II, Chapter II. Fractional powers, General roots of unity". Written at Dublin, Irland. In Hamilton, William Edwin (ed.). Elements of Quaternions (1 ed.). London, UK: Longmans, Green & Co., University Press, Michael Henry Gill. pp. 250–257, 260, 262–263. Retrieved 2016-01-17. pp. 250, 252: [...] cos [...] + i sin [...] we shall occasionally abridge to the following: [...] cis [...]. As to the marks [...], they are to be considered as chiefly available for the present exposition of the system, and as not often wanted, nor employed, in the subsequent practise thereof; and the same remark applies to the recent abrigdement cis, for cos + i sin [...] ([1], [2][3]) (NB. This work was published posthumously, Hamilton died in 1865.)

[Hamilton_1899-25] Hamilton, William Rowan (1899) [1866-01-01]. Hamilton, William Edwin; Joly, Charles Jasper (eds.). Elements of Quaternions. Vol. I (2 ed.). London, UK: Longmans, Green & Co. p. 262. Retrieved 2019-08-03. p. 262: [...] recent abridgment cis for cos + i sin [...] (NB. This edition was reprinted by Chelsea Publishing Company in 1969.)

[Stringham_1893-26] Stringham, Irving (1893-07-01) [1891]. Uniplanar Algebra, being part 1 of a propædeutic to the higher mathematical analysis. Vol. 1. C. A. Mordock & Co. (printer) (1 ed.). San Francisco, California, USA: The Berkeley Press. pp. 71–75, 77, 79–80, 82, 84–86, 89, 91–92, 94–95, 100–102, 116, 123, 128–129, 134–135. Retrieved 2016-01-18. p. 71: As an abbreviation for cos θ + i sin θ it is convenient to use cis θ, which may be read: sector of θ.

[Cajori_1929-27] Cajori, Florian (1952) [March 1929]. A History of Mathematical Notations. Vol. 2 (3rd corrected printing of 1929 issue, 2nd ed.). Chicago, Illinois, USA: Open court publishing company. p. 133. ISBN 978-1-60206-714-1. Retrieved 2016-01-18. p. 133: Stringham denoted cos β + i sin β by "cis β", a notation also used by Harkness and Morley. (NB. ISBN and link for reprint of 2nd edition by Cosimo, Inc., New York, US, 2013.)

[Harkness_Morley_1898-28] Harkness, James; Morley, Frank (1898). Introduction to the Theory of Analytic Functions (1 ed.). London, UK: Macmillan and Company. pp. 18, 22, 48, 52, 170. ISBN 978-1-16407019-1. Retrieved 2016-01-18. (NB. ISBN for reprint by Kessinger Publishing, 2010.)

[Campbell_1901-29] Campbell, George Ashley (1903) [1901-06-07]. "Chapter XXX. On loaded lines in telephonic transmission" (PDF). The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. Series 6. 5 (27). Taylor & Francis: 313–330. doi:10.1080/14786440309462928. Archived (PDF) from the original on 2023-07-16. Retrieved 2023-07-16. (2+18 pages)

[Campbell_1911-30] Campbell, George Ashley (April 1911). "Cisoidal oscillations" (PDF). Proceedings of the American Institute of Electrical Engineers. XXX (1–6). American Institute of Electrical Engineers: 789–824. doi:10.1109/PAIEE.1911.6659711. S2CID 51647814. Retrieved 2023-06-24. (37 pages)

[Campbell_1928-31] Campbell, George Ashley (1928-10-01) [1927-09-13]. "The Practical Application of the Fourier Integral" (PDF). The Bell System Technical Journal. 7 (4). American Telephone and Telegraph Company: 639–707 [641]. doi:10.1002/j.1538-7305.1928.tb00347.x. S2CID 53552671. Retrieved 2023-06-24. p. 641: It has been recognized, almost from the start, however, that the form which best combines mathematical simplicity and complete generality makes use of the exponential oscillating function e^i2πft. More recently the overwhelming advantage of using this oscillating function in the discussion of sinusoidal oscillatory systems has been generally recognized. It is, therefore, plain that this oscillating function should be adopted as the basic oscillation for both of the proposed tables. A name for this oscillation, associating it with sines and cosines, rather than with the real exponential function, seems desirable. The abbreviation cis x for (cos x + i sin x) suggests that we name this function a cis or a cisoidal oscillation. (69 pages)

[Hartley_1942-32] Hartley, Ralph V. L. (March 1942). "A More Symmetrical Fourier Analysis Applied to Transmission Problems". Proceedings of the IRE. 30 (3). Institute of Radio Engineers: 144–150. doi:10.1109/JRPROC.1942.234333. S2CID 51644127. Archived from the original on 2019-04-05. Retrieved 2023-07-16.

[Bracewell_1999-33] Bracewell, Ronald N. (June 1999) [1985, 1978, 1965]. The Fourier Transform and Its Applications (3 ed.). McGraw-Hill. ISBN 978-0-07303938-1.

[Diehl-Leupp_2010-34] Diehl, Christina; Leupp, Marcel (January 2010). Komplexe Zahlen: Ein Leitprogramm in Mathematik (PDF) (in German). Basel & Herisau, Switzerland: Eidgenössische Technische Hochschule Zürich (ETH). p. 41. Archived (PDF) from the original on 2017-08-27. p. 41: [...] Bitte vergessen Sie aber nicht, dass e^iφ für uns bisher nur eine Schreibweise für den Einheitszeiger mit Winkel φ ist. In anderen Büchern wird dafür oft der Ausdruck cis(φ) anstelle von e^iφ verwendet. [...] (109 pages)

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