cis函数

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cis函数示意图

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一个可以代表cis函数的图形，蓝色是实数部、橘色是虚数部
cis函数
File:Cis function.png
性质
奇偶性 N/A
定义域 (-∞,∞)
到达域 <math> \left
周期 2π
特定值
当x=0 1
当x=+∞ N/A
当x=-∞ N/A
最大值 复数无法比大小
最小值 复数无法比大小
其他性质
渐近线 N/A
根 N/A
临界点 N/A
拐点 kπ
不动点 0
k是一个整数.

在微积分学中，cis函数又称纯虚数指数函数，是复变函数的一种，和三角函数类似，其可以使用正弦函数和余弦函数<math>\operatorname{cis} x = \cos x + i \sin x</math>来定义，是一种实变数复数值函数（英语：Complex-valued function），其中<math>i</math>为虚数单位，而cis则为cos + i sin的缩写。

概观[编辑]

cis函数是欧拉公式等号右侧的所形的组合函数简写：

<math>e^{ix} = \cos x + i\sin x,</math>

其中i表示虚数单位<math>i^2=-1</math>。因此

<math>\operatorname{cis} x = \cos x + i\sin x,</math>[1][2][3]

cis符号最早由威廉·哈密顿在他于1866出版的《Elements of Quaternions》中使用[4]，而Irving Stringham在1893出版的《Uniplanar Algebra》 [5][6] 以及James Harkness和Frank Morley在1898出版的《Theory of Analytic Functions》中皆沿用了此一符号 [6][7] ，其利用欧拉公式将三角函数与复平面的指数函数连结起来。

cis函数主要的功能为简化某些数学表达式，透过cis函数可以使部分数学式能更简便地表达[4][5][8]，例如傅里叶变换和哈特利变换的结合[9][10][11]，以及应用在教学上时，因某些因素（如课程安排或课纲需求）因故不能使用指数来表达数学式时，cis函数就能派上用场。

性质[编辑]

cis函数的定义域是整个实数集，值域是单位复数，绝对值为1的复数。它是周期函数，其最小正周期为<math>2\pi</math>。其图像关于原点对称。

上述文字称它以类似三角函数的形式来定义函数的原因是，就如同三角函数，他也算是一种比值，复数和其模的比值:

<math>\operatorname{cis} \theta = \frac{z}{\left| z\right|}</math>，其中<math>z</math>是辐角为<math>\theta</math>的复数

因此，当一复数的模为1，其反函数就是辐角（arg函数）。

<math>\operatorname{cis}</math>函数可视为求单位复数的函数。

<math>\operatorname{cis}</math>函数的实数部分和余弦函数相同。

File:Cis function coloring plot 3D.png
cis函数 定义在复数。图中，颜色代表辐角，高代表模

微分[编辑]

<math>\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{cis} z = i\operatorname{cis} z = ie^{iz}</math>[1][12]

积分[编辑]

<math>\int\operatorname{cis} z \,\mathrm{d}z = -i\operatorname{cis} z = -ie^{iz}</math>[1]

其他性质[编辑]

根据欧拉公式，cis函数有以下性质：

<math>\operatorname{cis}(x+y) = \operatorname{cis} x\,\operatorname{cis} y</math>[13]
<math>\operatorname{cis}(x-y) = {\operatorname{cis} x \over \operatorname{cis} y}</math>

上述性质是当<math>x</math>与<math>y</math>都是复数时成立。在<math>x</math>与<math>y</math>都是实数时，有以下不等式：

<math>|\operatorname{cis} x - \operatorname{cis} y| \le |x-y|.</math>[13]

命名[编辑]

由于<math>\operatorname{cis}</math>函数的值为“余弦加上虚数单位倍的正弦”，取其英文缩写cosine and imaginary unit sine，故以<math>\operatorname{cis}</math>来表示该函数。

欧拉公式[编辑]

主条目：欧拉公式

在数学上，为了简化欧拉公式<math>e^{ix} = \cos x + i\sin x \ </math>，因此将欧拉公式以类似三角函数的形式来定义函数，给出了cis函数的定义[1][9][8][2][14][10][11][15]：

<math>\operatorname{cis} \theta = \cos \theta + i\;\sin \theta</math>

并且一般定义域为<math>\theta \in \mathbb{R}\,</math>，值域为<math>\theta \in \mathbb{C}\,</math>。

当<math>\theta</math>值为复数时，<math>\operatorname{cis}</math>函数仍然是有效的，因此可利用cis函数将欧拉公式推广到更复杂的版本。[16]

棣莫弗公式[编辑]

主条目：棣莫弗公式

在数学上，为了方便起见，可以将棣莫弗公式写成以下形式：

<math>\operatorname{cis}^n (x) = \operatorname{cis} (nx)</math>

指数定义[编辑]

跟其他三角函数类似，可以用e的指数来表示，依照欧拉公式给出: <math>\operatorname{cis} \theta = e^{i\theta}</math>

反函数[编辑]

<math>\operatorname{cis}</math>的反函数：<math>\operatorname{arccis} x</math>，当代入模为1的复数时，所得的值是其辐角

类似其他三角函数，<math>\operatorname{cis}</math>的反函数也可以用自然对数来表示

<math>\operatorname{arccis} \, x =\mathrm{i}} \ln x \,</math>

當一複數經過符號函數後代入<math>\operatorname{arccis} x</math>可得輻角。

恆等式[编辑]

<math>\operatorname{cis}</math>函數的倍角公式似乎比三角函數簡單許多

半形公式[编辑]

<math>\operatorname{cis} \frac{\theta}{2} = \frac{(1+i) + (1-i)\cos \theta}{\sin \theta}</math>
<math>\operatorname{cis} \frac{\theta}{2} = \sqrt{\operatorname{cis} \theta}</math>

倍角公式[编辑]

<math>\operatorname{cis} 2\theta = \operatorname{cis}^2 \theta</math>
<math>\operatorname{cis} n\theta = \operatorname{cis}^n \theta</math>

冪簡約公式[编辑]

<math>\operatorname{cis}^n \theta = \operatorname{cis} n\theta</math>

相關函數[编辑]

餘cis函數[编辑]

File:Cocis function.png
cocis函數，正好跟cis上下顛倒，周期相同，但是位移了<math>\frac{\pi}{2}</math>

就如同三角函數，我們可以令：<math>\operatorname{cocis} \theta = \cos \left(\frac{\pi}{2\theta\right) + i\;\sin \left(\frac{\pi}{2}-\theta\right) = \sin \theta + i\;\cos \theta</math>，其可用于诱导公式来化简某些特定的<math>\operatorname{cis}</math>函数的式子。

至于指数定义，经过正弦和余弦的指数定义得:

<math>\operatorname{cocis} \theta = \frac{(1-i)e^{i \theta }+(1+i)e^{-i \theta }}{2}</math>

有恒等式:

<math>\operatorname{cis} (-\theta) = -i\operatorname{cocis} \theta</math>
<math>\operatorname{cis} \left(\frac{\pi}{2}-\theta\right) = \operatorname{cocis} \theta</math>
<math>\operatorname{cis} \left(\frac{\pi}{2}+\theta\right) = i\operatorname{cis} \theta</math>
<math>\operatorname{cis} (\pi+\theta) = -\operatorname{cis} \theta</math>
<math>\operatorname{cocis} (-\theta) = i\operatorname{cis} \theta</math>
<math>\operatorname{cocis} \left(\frac{\pi}{2}-\theta \right) = \operatorname{cis} \theta</math>
<math>\operatorname{cocis} \left(\frac{\pi}{2}+\theta \right) = -i\operatorname{cocis} \theta</math>
<math>\operatorname{cocis} (\pi+\theta ) = -\operatorname{cocis} \theta</math>

双曲cis函数[编辑]

cish函数（<math>\cosh + i\sinh</math>）在几何意义上与cis函数对应的双曲函数不同。在双曲几何中，与欧几里得几何对应cis函数应为：

<math>e^{\theta}= \cosh(\theta) + \sinh(\theta)</math>

然而当中的<math>i</math>若定义为负一的平方根，则其会变为[17]：

<math>\operatorname{cish} \theta = \cosh(\theta) + i \sinh(\theta)</math>
双曲复数
主条目：双曲复数

在一般的情况下，cis函数对应的双曲函数定义域和值域皆为实数，但若定义双曲复数，考虑数<math>z=x+jy</math>，其中<math>x,y</math>是实数，而量<math>j</math>不是实数，但<math>j^2</math>是实数。选取<math>j^2=-1</math>，得到一般复数。取<math>+1</math>的话，便得到双曲复数。

而双曲复数有对应的欧拉公式:<math>e^{j \theta} = \cosh(\theta) + j \sinh(\theta)</math>

<math>\operatorname{cish} \theta = \cosh(\theta) + j \sinh(\theta)</math>

其中j为双曲复数。

因此双曲cis函数得到的值为双曲复数，相反的若将其反函数带入模为一的双曲复数可得其辐角。

如此一来，值域将会变成分裂四元数。

cas函数[编辑]

cas函数是一个以类似cis函数的概念定义的一个函数，为雷夫·赫特利（英语：Ralph Hartley）于1942提出，其定义为<math>\mathrm{cas}(x):=\cos x+\sin x</math>，是一种实变数实值函数，而cas为“cosine-and-sine”的缩写，其表示了实数值的赫特利变换（英语：Hartley transform）[18][19]：

<math>\mathrm{cas}(x)=\cos x+\sin x</math>

cas函数存在一些恒等式：

<math>

2 \operatorname{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). \, </math> 角和公式：

<math>

\operatorname{cas} (a+b) = {\cos (a) \operatorname{cas} (b)} + {\sin (a) \operatorname{cas} (-b)} = \cos (b) \operatorname{cas} (a) + \sin (b) \operatorname{cas}(-a) \, </math> 微分：

<math>

\operatorname{cas}'(a) = \frac{\mathrm{d}}{\mathrm{d}a} \operatorname{cas} (a) = \cos (a) - \sin (a) = \operatorname{cas}(-a). </math>

参见[编辑]

参考文献[编辑]

  1. ^ 1.0 1.1 1.2 1.3 Weisstein, Eric W. (编). Cis. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. [2016-01-09]. （原始内容存档于2016-01-27） （English）. 
  2. ^ 2.0 2.1 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）. 
  3. ^ Rationale for International Standard - Programming Languages - C (PDF). 5.10: 114, 117, 183, 186–187. April 2003 [2010-10-17]. （原始内容存档 (PDF)于2016-06-06）. 
  4. ^ 4.0 4.1 Hamilton, William Rowan. II. Fractional powers, General roots of unity. 写于Dublin. Hamilton, William Edwin (编). 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])
  5. ^ 5.0 5.1 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. 
  6. ^ 6.0 6.1 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.)
  7. ^ 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.)
  8. ^ 8.0 8.1 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. 
  9. ^ 9.0 9.1 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. Byrnes, Jim (编). 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）. 
  10. ^ 10.0 10.1 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. （原始内容存档于2018-10-17）. 
  11. ^ 11.0 11.1 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. （原始内容存档于2018-10-17）. 
  12. ^ Fuchs, Martin. Chapter 11: Differenzierbarkeit von Funktionen. Analysis I (PDF) WS 2011/2012. Fachrichtung 6.1 Mathematik, Universität des Saarlandes, Germany´. 2011: 3, 13 [2016-01-15]. （原始内容 (PDF)存档于2021-07-10） （Deutsch）. 
  13. ^ 13.0 13.1 Fuchs, Martin. Chapter 8.IV: Spezielle Funktionen – Die trigonometrischen Funktionen. Analysis I (PDF) WS 2011/2012. Fachrichtung 6.1 Mathematik, Universität des Saarlandes, Germany´. 2011: 16–20 [2016-01-15]. （原始内容 (PDF)存档于2021-01-20） （Deutsch）. 
  14. ^ 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）. 
  15. ^ 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）. 
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  17. ^ Ahangar, Reza. The Relativistic Geometry of the Complex Matter Space. Journal of Applied Mathematics and Physics. 2017-01, 05: 422–438. doi:10.4236/jamp.2017.52037. 
  18. ^ Hartley, Ralph V. L. A More Symmetrical Fourier Analysis Applied to Transmission Problems. Proceedings of the IRE. March 1942, 30 (3): 144–150 [2018-10-18]. doi:10.1109/JRPROC.1942.234333. （原始内容存档于2018-10-18）. 
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