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cis函數示意圖

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一個可以代表cis函數的圖形，藍色是實數部、橘色是虛數部。

cis函數

性質
奇偶性	N/A
定義域	(-∞,∞)
到達域	$left\|operatornamecisxright\| = 1 , ,operatornamecisx in mathbbC$
周期	2π
特定值
當x=0	1
當x=+∞	N/A
當x=-∞	N/A
最大值	複數無法比大小
最小值	複數無法比大小
其他性質
渐近线	N/A
根	N/A
臨界點	N/A
拐點	kπ
不動點	0
k是一個整數.

在初等解析學中，cis函數又稱純虛數指數函數，是複變函數的一种，和三角函數類似，其可以使用正弦函數和餘弦函數 $cis(x) : = cos(x) + i sin(x)$ 來定義，是一種實變數複數值函數（英语：Complex-valued function），其中 $i$ 為虛數單位，而cis則為 $c os + i s in$ 的縮寫。

概觀

cis符號最早由威廉·哈密頓在他於1866出版的《Elements of Quaternions》中使用^[1]，而Irving Stringham在1893出版的《Uniplanar Algebra》
^[2]^[3]
以及James Harkness和Frank Morley在1898出版的《Theory of Analytic Functions》中皆沿用了此一符號
^[3]^[4]
，其利用歐拉公式將三角函數與複平面的指數函數連結起來。

cis函數主要的功能為簡化某些數學表達式，透過cis函數可以使部分數學式能更簡便地表達^[1]^[2]^[5]，例如傅里葉變換和哈特利變換的結合^[6]^[7]^[8]，以及應用在教學上時，因某些因素（如課程安排或課綱需求）因故不能使用指數來表達數學式時，cis函數就能派上用場。

性質

cis函數的定义域是整个实数集，值域是單位複數，絕對值為1的複數。它是周期函数，其最小正周期为 $2pi$ 。其图像关于原点对称。

上述文字稱它以類似三角函數的形式來定義函數的原因是，就如同三角函數，他也算是一種比值，複數和其模的比值:

operatornamecis theta = fracz zright

，其中

z

是幅角為

theta

的複數

因此，當一複數的模為1，其反函數就是幅角（arg函數）。

$displaystyle operatorname cis$ 函數可視為求單位複數的函數

$displaystyle operatorname cis$ 函數的實數部分和餘弦函數相同。

cis函數定義在複數。圖中，顏色代表幅角，高代表模。

命名

由於 $displaystyle operatorname cis$ 函數的值為「餘弦加上虛數單位倍的正弦」，取其英文縮寫cosine and imaginary unit sine，故以 $displaystyle operatorname cis$ 來表示該函數。

歐拉公式

在數學上，為了簡化歐拉公式 $e^ix = cos x + isin x$ ，因此將歐拉公式以類似三角函數的形式來定義函數，給出了cis函數的定義^[9]^[6]^[5]^[10]^[11]^[7]^[8]^[12]：

operatornamecis theta = cos theta + i;sin theta

並且一般定義域為 $theta in mathbbR,$ ，值域為 $theta in mathbbC,$ 。

當 $theta$ 值為複數時， $displaystyle operatorname cis$ 函數仍然是有效的，所以有些人可利用cis函數將歐拉公式推廣到更複雜的版本。^[13]

指數定義

跟其他三角函數類似，可以用e的指數來表示，依照歐拉公式給出:
$operatornamecis theta = e^itheta$

反函數

$displaystyle operatorname cis$ 的反函數: $displaystyle operatorname arccis x$ ，當代入模為1的複數時，所得的值是其輻角

類似其他三角函數， $displaystyle operatorname cis$ 的反函數也可以用自然對數來表示

operatornamearccis , x =-mathrmi ln x ,

當一複數經過符號函數後代入 $displaystyle operatorname arccis x$ 可得輻角。

恆等式

$displaystyle operatorname cis$ 函數的倍角公式似乎比三角函數簡單許多

半形公式

operatornamecis fractheta2 = frac(1+i) + (1-i)cos thetasin theta

operatornamecis fractheta2 = sqrtoperatornamecis theta

倍角公式

operatornamecis 2theta = operatornamecis^2 theta

operatornamecis ntheta = operatornamecis^n theta

冪簡約公式

operatornamecis^n theta = operatornamecis ntheta

相關函數

餘cis函數

cocis函數，正好跟cis上下顛倒，周期相同，但是位移了

fracpi2

就如同三角函數，我們可以令： $operatorname cocistheta =cos left(frac pi 2-theta right)+i;sin left(frac pi 2-theta right)=sin theta +i;cos theta$ ，其可用於誘導公式來化簡某些特定的 $displaystyle operatorname cis$ 函數的式子。

至於指數定義，經過正弦和餘弦的指數定義得:

operatornamecocis theta = frac(1-i)e^i theta +(1+i)e^-i theta 2

有恆等式:

operatornamecis (-theta) = -ioperatornamecocis theta

operatorname cisleft(frac pi 2-theta right)=operatorname cocistheta

operatorname cisleft(frac pi 2+theta right)=ioperatorname cistheta

operatornamecis (pi+theta) = -operatornamecis theta

operatornamecocis (-theta) = ioperatornamecis theta

operatorname cocisleft(frac pi 2-theta right)=operatorname cistheta

operatorname cocisleft(frac pi 2+theta right)=-ioperatorname cocistheta

operatornamecocis (pi+theta ) = -operatornamecocis theta

雙曲cis函數

一般會將雙曲cis函數定義成:

operatornamecish theta = e^theta= cosh(theta) + sinh(theta)

定義域和值域皆為實數，但若定義雙曲複數，

考慮數 $z=x+jy$ ，其中 $x,y$ 是實數，而量 $j$ 不是實數，但 $j^2$ 是實數。

選取 $j^2=-1$ ，得到一般複數。取 $+1$ 的話，便得到雙曲複數。

而雙曲複數有對應的歐拉公式: $e^j theta = cosh(theta) + j sinh(theta)$

operatornamecish theta = cosh(theta) + j sinh(theta)

其中j為雙曲複數。

因此雙曲cis函數得到的值為雙曲複數，相反的若將其反函數帶入模為一的雙曲複數可得其輻角。

如此一來，值域將會變成四元數。

cas函數

cas函數是一個以類似cis函數的概念定義的一個函數，為雷夫·赫特利（英语：Ralph Hartley）於1942提出，其定義為 $cas(x) : = cos(x) + sin(x)$ ，是一種實變數實值函數，而cas為「cosine-and-sine」的縮寫，其表示了實數值的赫特利變換（英语：Hartley transform）^[14]^[15]：

cas(x) = cos(x) + sin(x)

cas函數存在一些恆等式：

displaystyle 2operatorname cas (a+b)=operatorname cas (a)operatorname cas (b)+operatorname cas (-a)operatorname cas (b)+operatorname cas (a)operatorname cas (-b)-operatorname cas (-a)operatorname cas (-b).,

角和公式：

displaystyle operatorname cas (a+b)=cos(a)operatorname cas (b)+sin(a)operatorname cas (-b)=cos(b)operatorname cas (a)+sin(b)operatorname cas (-a),

微分：

displaystyle operatorname cas '(a)=frac mathrm d mathrm d aoperatorname cas (a)=cos(a)-sin(a)=operatorname cas (-a).

參見

正弦

餘弦

複數 (數學)

三角函数

三角函数恆等式

歐拉公式

參考文獻

^ ^1.0^1.1 Hamilton, William Rowan. II. Fractional powers, General roots of unity. 写于Dublin. (编) Hamilton, William Edwin. Elements of Quaternions. University Press, Michael Henry Gill, Dublin (printer) 1. London, UK: Longmans, Green & Co. 1866-01-01: 250–257, 260, 262–263 [2016-01-17]. […] 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])

^ ^2.0^2.1 Stringham, Irving. Uniplanar Algebra, being part 1 of a propædeutic to the higher mathematical analysis 1. C. A. Mordock & Co. (printer) 1. San Francisco, US: The Berkeley Press. 1893-07-01: 71–75, 77, 79–80, 82, 84–86, 89, 91–92, 94–95, 100–102, 116, 123, 128–129, 134–135 [1891] [2016-01-18]. As an abbreviation for cos θ + i sin θ it is convenient to use cis θ, which may be read: sector of θ.

^ ^3.0^3.1 Cajori, Florian. A History of Mathematical Notations 2 2 (3rd corrected printing of 1929 issue). Chicago, US: Open court publishing company. 1952: 133 [March 1929] [2016-01-18]. ISBN 978-1-60206-714-1. ISBN 1-60206-714-7. 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. Introduction to the Theory of Analytic Functions 1. London, UK: Macmillan and Company. 1898: 18, 22, 48, 52, 170 [2016-01-18]. ISBN 978-1-16407019-1. ISBN 1-16407019-3. (NB. ISBN for reprint by Kessinger Publishing, 2010.)

^ ^5.0^5.1 Swokowski, Earl; Cole, Jeffery. Precalculus: Functions and Graphs. Precalculus Series 12 (Cengage Learning). 2011 [2016-01-18]. ISBN 978-0-84006857-6. ISBN 0-84006857-3.

^ ^6.0^6.1 L.-Rundblad, Ekaterina; Maidan, Alexei; Novak, Peter; Labunets, Valeriy. Fast Color Wavelet-Haar-Hartley-Prometheus Transforms for Image Processing. 写于Prometheus Inc., Newport, USA. (编) Byrnes, Jim. Computational Noncommutative Algebra and Applications (PDF). NATO Science Series II: Mathematics, Physics and Chemistry (NAII) 136. Dordrecht, Netherlands: Springer Science + Business Media, Inc. 2004: 401-411 [2017-10-28]. ISBN 978-1-4020-1982-1. ISSN 1568-2609. doi:10.1007/1-4020-2307-3. （原始内容存档 (PDF)于2017-10-28）.

^ ^7.0^7.1 Kammler, David W. A First Course in Fourier Analysis 2. Cambridge University Press. 2008-01-17 [2017-10-28]. ISBN 978-1-13946903-6. ISBN 1-13946903-7.

^ ^8.0^8.1 Lorenzo, Carl F.; Hartley, Tom T. The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science. John Wiley & Sons. 2016-11-14 [2017-10-28]. ISBN 978-1-11913942-3. ISBN 1-11913942-2.

^ Weisstein, Eric W. Cis. MathWorld. Wolfram Research, Inc. 2015 [2000] [2016-01-09]. （原始内容存档于2016-01-27）.

^ Simmons, Bruce. Cis. Mathwords: Terms and Formulas from Algebra I to Calculus. Oregon City, OR, US: Clackamas Community College, Mathematics Department. 2014-07-28 [2004] [2016-01-15]. （原始内容存档于2016-01-19）.

^ Simmons, Bruce. Polar Form of a Complex Number. Mathwords: Terms and Formulas from Algebra I to Calculus. Oregon City, OR, US: Clackamas Community College, Mathematics Department. 2014-07-28 [2004] [2016-01-15]. （原始内容存档于2016-01-23）.

^ Pierce, Rod. Complex Number Multiplication. Maths Is Fun. 2016-01-04 [2000] [2016-01-15]. （原始内容存档于2016-01-15）.

^ Moskowitz, Martin A. A Course in Complex Analysis in One Variable. World Scientific Publishing Co. 2002: 7. ISBN 981-02-4780-X.

^ Hartley, Ralph V. L. A More Symmetrical Fourier Analysis Applied to Transmission Problems. Proceedings of the IRE. March 1942, 30 (3): 144–150. doi:10.1109/JRPROC.1942.234333.

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

[Hamilton_1866-1] 1.0^1.1 Hamilton, William Rowan. II. Fractional powers, General roots of unity. 写于Dublin. (编) Hamilton, William Edwin. Elements of Quaternions. University Press, Michael Henry Gill, Dublin (printer) 1. London, UK: Longmans, Green & Co. 1866-01-01: 250–257, 260, 262–263 [2016-01-17]. […] 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])

[Stringham_1893-2] 2.0^2.1 Stringham, Irving. Uniplanar Algebra, being part 1 of a propædeutic to the higher mathematical analysis 1. C. A. Mordock & Co. (printer) 1. San Francisco, US: The Berkeley Press. 1893-07-01: 71–75, 77, 79–80, 82, 84–86, 89, 91–92, 94–95, 100–102, 116, 123, 128–129, 134–135 [1891] [2016-01-18]. As an abbreviation for cos θ + i sin θ it is convenient to use cis θ, which may be read: sector of θ.

[Cajori_1929-3] 3.0^3.1 Cajori, Florian. A History of Mathematical Notations 2 2 (3rd corrected printing of 1929 issue). Chicago, US: Open court publishing company. 1952: 133 [March 1929] [2016-01-18]. ISBN 978-1-60206-714-1. ISBN 1-60206-714-7. 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_1898-4] Harkness, James; Morley, Frank. Introduction to the Theory of Analytic Functions 1. London, UK: Macmillan and Company. 1898: 18, 22, 48, 52, 170 [2016-01-18]. ISBN 978-1-16407019-1. ISBN 1-16407019-3. (NB. ISBN for reprint by Kessinger Publishing, 2010.)

[Swokowski_2011-5] 5.0^5.1 Swokowski, Earl; Cole, Jeffery. Precalculus: Functions and Graphs. Precalculus Series 12 (Cengage Learning). 2011 [2016-01-18]. ISBN 978-0-84006857-6. ISBN 0-84006857-3.

[Byrnes_2004-6] 6.0^6.1 L.-Rundblad, Ekaterina; Maidan, Alexei; Novak, Peter; Labunets, Valeriy. Fast Color Wavelet-Haar-Hartley-Prometheus Transforms for Image Processing. 写于Prometheus Inc., Newport, USA. (编) Byrnes, Jim. Computational Noncommutative Algebra and Applications (PDF). NATO Science Series II: Mathematics, Physics and Chemistry (NAII) 136. Dordrecht, Netherlands: Springer Science + Business Media, Inc. 2004: 401-411 [2017-10-28]. ISBN 978-1-4020-1982-1. ISSN 1568-2609. doi:10.1007/1-4020-2307-3. （原始内容存档 (PDF)于2017-10-28）.

[Kammler_2008-7] 7.0^7.1 Kammler, David W. A First Course in Fourier Analysis 2. Cambridge University Press. 2008-01-17 [2017-10-28]. ISBN 978-1-13946903-6. ISBN 1-13946903-7.

[Lorenzo-Hartley_2016-8] 8.0^8.1 Lorenzo, Carl F.; Hartley, Tom T. The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science. John Wiley & Sons. 2016-11-14 [2017-10-28]. ISBN 978-1-11913942-3. ISBN 1-11913942-2.

[Weisstein_cis-9] Weisstein, Eric W. Cis. MathWorld. Wolfram Research, Inc. 2015 [2000] [2016-01-09]. （原始内容存档于2016-01-27）.

[Simmons_2014_1-10] Simmons, Bruce. Cis. Mathwords: Terms and Formulas from Algebra I to Calculus. Oregon City, OR, US: Clackamas Community College, Mathematics Department. 2014-07-28 [2004] [2016-01-15]. （原始内容存档于2016-01-19）.

[Simmons_2014_2-11] Simmons, Bruce. Polar Form of a Complex Number. Mathwords: Terms and Formulas from Algebra I to Calculus. Oregon City, OR, US: Clackamas Community College, Mathematics Department. 2014-07-28 [2004] [2016-01-15]. （原始内容存档于2016-01-23）.

[Pierce_2016-12] Pierce, Rod. Complex Number Multiplication. Maths Is Fun. 2016-01-04 [2000] [2016-01-15]. （原始内容存档于2016-01-15）.

[13] Moskowitz, Martin A. A Course in Complex Analysis in One Variable. World Scientific Publishing Co. 2002: 7. ISBN 981-02-4780-X.

[Hartley_1942-14] Hartley, Ralph V. L. A More Symmetrical Fourier Analysis Applied to Transmission Problems. Proceedings of the IRE. March 1942, 30 (3): 144–150. doi:10.1109/JRPROC.1942.234333.

[Bracewell_1999-15] Bracewell, Ronald N. The Fourier Transform and Its Applications 3. McGraw-Hill. June 1999 [1985, 1978, 1965]. ISBN 978-0-07303938-1.

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