5.7 Complex Numbers and Functions: Glossary

Costas Loizou; Shuwei Chen; Deak Kirkham

5.7 Complex Numbers and Functions: Glossary

The pronunciation guide given in these glossaries draws on the symbols International Phonetic Alphabet. We have made two adaptations to the IPA system. Firstly, we use slanted lines as opposed to square brackets to enclose pronunciation material. We do this as the slanted line convention is more common in language teaching contexts. Secondly, we use the symbol <r> to denote the voiced alveolar approximant sound which appears in British English at the beginning of the each word in the phrase 'red rhombus'; the IPA would instead use an inverted 'r' symbols namely <ɹ>. We have done this to avoid unnecessary complication to the visuals of the glossaries.

In addition to these conventions, please also note that the pronunciations given are indicative of contemporary Standard British English pronunciation and should not be read as normative, nor as a target pronunciation that you must aim to produce. The pronunciation of English varies widely across World Englishes; part of becoming fluent in the academic English of mathematics is developing an awareness of, and sensitivity to, the variation in pronunciation that you will encounter.

Word	Pronunciation	Part of speech词性	Related terminology	Chinese meaning	Example sentence 1	Example sentence 2
complex number	/ˈkɒmpleks ˈnʌmbə/	n. phr.	real number 实数	复数	A complex number \(z\) is real if and only if \(\Im (z)=0\). 复数 \(z\) 为实数当且仅当 \(\Im (z)=0\)。	Dividing a complex number often uses its conjugate to simplify the denominator. 复数除法常利用共轭复数使分母实数化。
imaginary number	/ɪˈmædʒɪnəri ˈnʌmbə/	n. phr.	imaginary part 虚部； imaginary unit 虚数单位	虚数	An imaginary number is a special case of a complex number where the real part is zero. The imaginary unit satisfies \(i^2=−1\), which defines pure imaginary numbers like \(2i\), \(−5i\). 虚数单位满足 \(i^2=−1\)，由此定义纯虚数如 \(2i\), \(−5i\)。	For \(z=0+6i\), we say \(z\) is a purely imaginary complex number. 若 \(z=0+6i\)，称 \(z\) 为纯虚复数。
Argand plane	/ˈɑːɡænd pleɪn/	n. phr.	complex plane 复平面（同阿尔冈平面）； Argand diagram 阿尔冈图	复平面	In the Argand plane, \(z=x+yi\) corresponds to the point \((x,y)\). 在阿尔冈平面中，\(z=x+yi\) 对应点 \((x,y)\)。	Multiplication by \(i\) rotates points by \(\frac{\pi}{2}\) in the Argand plane. 乘以 \(i\) 在阿尔冈平面中旋转 \(\frac{\pi}{2}\)。
representation	/ˌreprɪzenˈteɪʃ(ə)n/	n.		表示	The exponential representation \(z=r e^{i \theta}\) comes directly from Euler’s formula. 指数表示 \(z=r e^{i \theta}\) 直接来自欧拉公式。	Every representation satisfies \(z=\Re (z)+i \Im (z)\). 任何表示都满足 \(z=\Re (z)+ i \Im (z)\)。
rectangular form	/rekˈtæŋɡjʊlə fɔːm/	n. phr.	Cartesian form 笛卡尔坐标形式（同直角坐标形式）	直角坐标形式	For \(z=2 e^{i \frac{\pi}{3}}\), rectangular form is \(z=1+i3\). 对\(z=2 e^{i \frac{\pi}{3}}\)，直角坐标形式为 \(z=1+i3\)。	Addition is easiest in rectangular form: \((x_1+x_2)+(y_1+y_2)i\). 加法在直角坐标形式下最简单：\((x_1+x_2)+(y_1+y_2)i\)。
polar form	/ˈpəʊlə fɔːm/	n. phr.	polar coordinates 极坐标	极坐标形式	The polar form of \(z=1+i\) is \(z=2(\cos (\frac{\pi}{4}) + i \sin (\frac{\pi}{4}))\). \(z=1+i\) 的极坐标形式为 \(z=2(\cos (\frac{\pi}{4}) + i \sin (\frac{\pi}{4}))\)。	In polar form, \(z=r e^{i \theta}\) where \(r=\|z\|\) and \(\theta=\arg (z)\). 极坐标形式中 \(z=r e^{i \theta}\)，其中 \(r=\|z\|\), \(\theta=\arg (z)\)。
exponential form	/ˌekspəˈnenʃəl fɔːm/	n. phr.		指数形式	For \(z=\cos \theta + i \sin \theta\), exponential form is \(z=e^{i \theta}\). 复数 \(z=\cos \theta + i \sin \theta\) 的指数形式为 \(z=e^{i \theta}\)。	A complex number can be written in exponential form as \(z=r e^{i \theta}\). 复数可以写成指数形式\(z=r e^{i \theta}\)。
modulus	/ˈmɒdjʊləs/	n.		模（数学术语）	Let \(z\) be a complex number with modulus \(1\). 设复数 \(z\) 的模为 \(1\)。	The modulus of \(z= a + bi\) is given by \(\|z\|\). 复数的模用 \(\|z\|\) 表示。
magnitude	/ˈmæɡnɪtjuːd/	n.		大小，模（适用于物理量，向量）	The magnitude of a vector represents its length. 向量的大小表示它的长度。	The magnitude of the complex number is the distance from the origin. 复数的大小是它到原点的距离。
argument	/ˈɑːɡjʊmənt/	n.		幅角（复数在复平面上所对应的向量与正实轴之间的夹角）；论证；自变量，元	In polar form a complex number is written using its modulus and argument. 在极坐标形式中复数用模和辐角表示。	The argument of \(z=x+iy\) is denoted by \(\arg (z)\). 复数\(z=x+iy\) 的辐角记作 \(\arg (z)\)。
complex conjugate	/ˈkɒmpleks ˈkɒndʒʊɡət/	n. phr.		共轭复数	The complex conjugate of \(z=a+bi\) is \(a-bi\). 复数\(z=a+bi\)的共轭是\(a-bi\)。	The product of a complex number and its complex conjugate is \(\|z\|^2\). 复数与其共轭的乘积等于模的平方。
Euler’s formula	/ˈɔɪləz ˈfɔːmjʊlə/	n. phr.	Euler’s identity 欧拉恒等式	欧拉公式	Using Euler’s formula, we derive \(\cos \theta = \frac{e^{i \theta}+e^{-i \theta}}{2}\). 由欧拉公式得 \(\cos \theta = \frac{e^{i \theta}+e^{-i \theta}}{2}\)。	Euler's formula states that \(e^{i \theta}=\cos \theta + i \sin \theta\). 欧拉公式表明\(e^{i \theta}=\cos \theta + i \sin \theta\)。
De Moivre’s theorem	/də ˈmwɑːvz ˈθɪərəm/	n. phr.		棣莫弗定理	We use De Moivre’s theorem to compute \((\cos \theta + i \sin \theta)^5\) efficiently. 我们用棣莫弗定理高效计算\((\cos \theta + i \sin \theta)^5\)。	To find \((2(\cos \theta + i \sin \theta))^n\) we apply De Moivre’s theorem. 要计算\((2(\cos \theta + i \sin \theta))^n\)可以使用棣莫弗定理。
trigonometric form	/ˌtrɪɡənəˈmetrɪk fɔːm/	n. phr.		三角形式	We use trigonometric form to compute powers like \(z^n = r^n(\cos (n \theta) + i \sin (n \theta)\). 用三角形式计算幂 \(z^n = r^n(\cos (n \theta) + i \sin (n \theta)\)。	In trigonometric form multiplying complex numbers involves multiplying moduli and adding arguments. 在三角形式中复数相乘是模相乘辐角相加。
analytic function	/ˌænəˈlɪtɪk ˈfʌŋkʃən/	n. phr.		解析函数	An analytic function is differentiable at every point in its domain. 解析函数在其定义域内处处可导。	Every polynomial is an analytic function. 每个多项式都是解析函数。
singularity	/ˌsɪŋɡjʊˈlærɪti/	n.		奇点	A singularity is a point where the function is not analytic. 奇点是函数不解析的点。	The function \(f(z)=\frac{1}{z}\) has a singularity at \(z=0\). 函数\(f(z)=\frac{1}{z}\)在\(z=0\)处有奇点。
closed under	/kləʊzd ˈʌndə/	phr.		对......封闭	The set of integers is closed under addition. 整数集对加法是封闭的。	Complex numbers are closed under division by non-zero complex numbers. 复数在除以非零复数时是封闭的。
qualitatively	/ˈkwɒlɪteɪtɪvli/	adv.	qualitative, adj.	定性地	Qualitatively, \(\|z\|\) measures distance from \(0\) in the complex plane. 定性地说，\(\|z\|\) 衡量复平面上到 \(0\) 的距离。	Near a singularity, a function \(f(z)\) behaves qualitatively like \(\frac{1}{(z−a)^n}\). 在奇点附近，函数 f(z) 定性表现为 \(\frac{1}{(z−a)^n}\) 形式。
predict	/prɪˈdɪkt/	v.		预测	Using De Moivre’s theorem, we predict \((\cos (\frac{\pi}{4})+ i \sin (\frac{\pi}{4}))^8=1\). 用棣莫弗定理可预测 \((\cos (\frac{\pi}{4})+ i \sin (\frac{\pi}{4}))^8=1\)。	So there we can see, we basically predicted this answer in advance before...From symmetry, we predict all \(n\)-th roots satisfy \(\|z\|=1\). 由对称性可预测所有 \(n\) 次根满足 \(\|z\|=1\)。
exclude	/ɪkˈskluːd/	v.		排除	In defining \(\arg (z)\), we exclude negative values of r by requiring \(r=\|z\| \gt 0\). 定义 \(\arg (z)\) 时，通过 \(r=\|z\| \gt 0\) 排除负半径。	We exclude \(z=2\) from the domain because \(f(z)\) is not analytic there. 我们从定义域中排除 \(z=2\)，因 \(f(z)\) 在此处不解析。
corresponding	/ˌkɒrɪˈspɒndɪŋ/	adj.		相应的	For each value of \(x\) there is a corresponding value of \(y\). 对每个\(x\)值都有一个对应的\(y\)值。	The point \((x,y)\) has a corresponding point in the polar coordinate system. 点\((x,y)\)在极坐标系中有一个对应点。
bridge	/brɪdʒ/	v.		联系，连接（数学结构之间的）	Euler’s formula is a key identity bridging exponential and trigonometric functions. 欧拉公式是连接指数函数与三角函数的关键恒等式。	De Moivre’s theorem provides a tool bridging trigonometric expressions and powers of complex numbers. 棣莫弗定理提供了一种连接三角表达式与复数幂的工具。
uniformly spaced	/ˈjuːnɪfɔːmli speɪst/	adj. phr.		均匀分布	The \(n\)th roots of a complex number are uniformly spaced on a circle. 复数的 \(n\) 次方根在圆上均匀分布。	The arguments of the roots are uniformly spaced by \(\frac{2 \pi}{n}\). 各根的辐角以\(\frac{2 \pi}{n}\)为间隔均匀分布。
indeterminate	/ˌɪndɪˈtɜːmɪnət/	adj. phr.		未定义的	The argument of a complex number at \(z=0\) is indeterminate. 复数在\(z=0\)时的辐角未定义。	The value of the expression \(\frac{0}{0}\) is indeterminate. 表达式\(\frac{0}{0}\)是不定式
hyperbolic functions	/ˌhaɪpəˈbɒlɪk ˈfʌŋkʃənz/	n. phr.	hyperbolic adj. 双曲线的； hyperbolic sine (sinh) 双曲正弦； hyperbolic cosine (cosh) 双曲余弦； hyperbolic tangent (tanh) 双曲正切	双曲函数	The identity \(\cosh^2 x - \sinh^2 x = 1\) is fundamental for hyperbolic functions. 恒等式\(\cosh^2 x - \sinh^2 x = 1\)是双曲函数的基本性质。	Hyperbolic functions are closely related to Euler’s formula in complex analysis. 双曲函数与复分析中的欧拉公式密切相关。