从三次方程到复平面：复数概念的奇妙演进（四）

注：本文为 “复数 | 历史 / 演进” 相关文章合辑。

因 csdn 篇幅限制分篇连载，此为第四篇。

生料，不同的文章不同的点。

机翻，未校。

Complex number and its discovery history

复数及其发现历史

Wenhao Chen¹, †, Dazheng Zhang², † and Yuteng Zou³, †,

¹SMIC High School, Beijing, China

²Wuhan Britain-China School, Wuhan, China

³Shenzhen College of International Education, Shenzhen, China

Abstract

摘要

The operations of complex numbers are the main subject of the mathematical analysis area known as complex analysis. It is also known as the theory of functions of a complex variable. The primary research topic in the field of complex analysis is holomorphic functions. These functions are defined on the complex plane, have differentiable properties, and allow for negative values. The residue theorem, the Cauchy integral formula, the Laurent series expansion, etc. are a few concepts, ideas, and methods that are commonly used in research. In particular, over the years, complex analysis in mathematics, physics, and engineering has been extensively used in algebraic geometry, fluid dynamics, quantum mechanics, and other related areas. Two Italian mathematicians, Girolamo Cardano and Raphael Bombelli, discovered complex numbers in the 16th century while attempting to solve a particular algebra, and Cauchy and Riemann extended it in the 19th century. This essay begin with investigation of arithmetic propertity of comlex numbers and then fully explores history development of complex numbers.

复数运算是数学分析领域中被称为复分析的主要研究对象。它也被称为复变函数论。复分析领域的主要研究课题是全纯函数。这些函数定义在复平面上，具有可微性，并且可以取负值。留数定理、柯西积分公式、洛朗级数展开等是研究中常用的一些概念、思想和方法。特别是多年来，复分析在数学、物理和工程领域被广泛应用于代数几何、流体动力学、量子力学等相关领域。16 世纪，两位意大利数学家吉罗拉莫・卡尔达诺（Girolamo Cardano）和拉斐尔・邦贝利（Raphael Bombelli）在试图求解特定代数问题时发现了复数，19 世纪柯西（Cauchy）和黎曼（Riemann）对其进行了拓展。本文从研究复数的算术性质入手，全面探讨复数的历史发展。

Keywords: Complex Plane, Cubic equation, Complex numbers.

关键词：复平面；三次方程；复数

1.Introduction

引言

The set of complex numbers $\mathbb{C}:=x+iy:x,y\in \mathbb{R}$ where $i=\sqrt{-1}(i^2=-1)$. The coordinate plane ${\mathbb{R}}^2$ can be consider as the visualized form of $\mathbb{C}$ where coordinate pair $(x,y)$ plotted as $z=x+iy$. On this occasion, real axis and the imaginary axis are $x$ -axis and $y$ -axis. This coordinate plane is complex plane. According to the distance formula, in the complex plane, $|z|$ can be considered as the distance from $z$ to $o$. Moreover, the distance from $z$ to $w$ is $|z-w|$. The addition can be visualized in complex plane as vector space addition on ${\mathbb{R}}^2$. Besides, the way to visualize multiplication is more complex. Because the polar coordinates can represent every point in ${\mathbb{R}}^2$. For any $z\in \mathbb{C}(z\ne 0)$, $|\frac{z}{|z|}|=|z|\cdot |\frac{1}{|z|}|=|z|\cdot \frac{1}{|z|}=1$. As a result, $\theta \in \mathbb{R}$ exists and $\frac{z}{|z|}=\cos \theta +i\sin \theta$. As write $e^{i\theta }=\cos \theta +i\sin \theta$. Name of the equation $z=|z|e^{i\theta }$ is the polar form of $z$ for $z\in \mathbb{C}$. The argument of $z$ is the name of the angle $\theta \in \mathbb{R}$, which is denoted $\arg z$. Trigonometric identities can be used to check that $e^{i{\theta }_1}{e}^{i{\theta }_2}=e^{i({\theta }_1+{\theta }_2)}$. Therefore, for $z,w\in \mathbb{C}$, $zw=|z|e^{i\arg z}\cdot |w|e^{i\arg w}=|zw|\cdot e^{i(\arg z+\arg w)}$. Thus, $|w|$ scales $z$ and $\arg w$ rotates counterclockwise compose $z\to zw$. Moreover, counterclockwise rotation by $(i=\frac{\pi }{2})$ is corresponds to $z\to iz$. Eventually, the real axis reflection is corresponded to complex conjugation $\overline{x+iy}=x-iy$. Since distance can be obtained by absolute value in complex planes, triangular inequality can be noted as $|a+b|\le |a|+|b|$. From the perspective of inverse triangular inequality, it could be written as $-||a|-|b||\le |a-b|$ and it can also lead to that $|a|=|a-b+b|\le |a-b|+|b|$. From these inequalities, they will result in the following $|a|-|b|\le |a-b|$, $|b|=|b-a+a|\le |b-a|+|a|$, $-(|a|-|b|)=|b|-|a|\le |b-a|=|a-b|$. Most importantly, it get $|\text{Im}(a)|\le |a|$ and $|\text{Re}(a)|\le |a|$. Convergence is generally defined as a sequence $(W_n{)}_{n\in \mathbb{N}}\subset \mathbb{C}$ converges to $Z\in \mathbb{C}$ if ${\lim }_{n\to \infty }(W_n-Z)=0$ which can be noted as ${\lim }_{n\to \infty }{W}_n=Z$ where $z$ was known as the limit of the sequence. Briefly, it can be written as $(W_n{)}_{n\in \mathbb{N}}$ converges to $Z$ for ${\forall }_n\ge n$, $|W_n-Z|<C$. As $|\text{Im}(a)|\le |a|$, $|\text{Re}(a)|\le |a|$ and $|a|=(\text{Re}(a{)}^2+\text{Im}(a{)}^2{)}^{\frac{1}{2}}$. For, ${\lim }_{n\to \infty }{W}_n=Z\to {\lim }_{n\to \infty }\text{Re}(W_n)={\lim }_{n\to \infty }\text{Im}(W_n)=Z$. So, the notion of convergence agrees with the virtual one on ${\mathbb{R}}^2$. $W_n\in \mathbb{C}$, a Cauchy sequence when satisfies ${\forall }_{\epsilon }>0$ and ${\exists }_N\in \mathbb{N}$, gives that $|W_n-W_m|<\epsilon$ in ${\forall }_{n,m}\ge N$. Thus, terms in Cauchy sequence can be conclude such that they would end up like what one person would expect them to be. For $P_0\in \mathbb{C}$ and $r>0$, where $r$ is the radius of an open disc centered at $P_0$, the disc can be noted as $P_r(P_0):=P\in \mathbb{C}:|P-P_0|<r$. Thus, such a closed disc can be written as: $\overline{P_r}(P_0):=P\in \mathbb{C}:|P-P_0|\le r$. And the circle with radius $r$ would be: $C_r(P_0)$ sign the radius $r$ or the center $P_0$. Then it enables us to only keep the following part as the notation Here’s a handy way to see if the offer is over. Convergence of a sequence of complex numbers is for the unit disk: D : = D 1 ( 0 ) = { P ∈ C : ∣ P ∣ < 1 } D:=D_{1}(0)=\{P \in \mathbb {C}:|P|<1\} D:=D1​(0)={P∈C:∣P∣<1}. Next, the focus turns to the topological properties of $\mathbb{C}$. Since the complex plane can be identified by ${\mathbb{R}}^2$ and has the same concept of distance and convergence, these properties are just a transformation from ${\mathbb{R}}^2$ to $\mathbb{C}$. Most importantly, as can be seen from the examples above, this is not a dichotomy between open and closed: the whole is open or closed, or both. Usually, you can tell whether a set is open or not directly from the definition. $:=P\in \mathbb{C}:|P-P_0|<r$. When any of the above is not important to the discussion, people can defined in the same way as it is for real numbers. The necessary and sufficient condition for the sequence { Z n } \{Z_{n}\} {Zn​} to converge to $\omega$ is that the real and imaginary parts converge to the real and imaginary parts of $\omega$, respectively. For all Cauchy sequences { Z n } \{Z_{n}\} {Zn​}, when $n,m\to +\infty$, it holds that $|Z_n-Z_m|\to 0$. The set of real numbers is complete, the Cauchy sequence of real numbers converges, and the convergence in the range of complex numbers is equivalent to the convergence of two Cauchy sequence of real numbers. Thus the set of complex numbers is complete.

复数集 C : = { x + i y : x , y ∈ R } \mathbb {C}:=\{x + iy: x, y \in \mathbb {R}\} C:={x+iy:x,y∈R}，其中 $i=\sqrt{-1}(i^2=-1)$。坐标平面 ${\mathbb{R}}^2$ 可以看作是 $\mathbb{C}$ 的可视化形式，坐标对 $(x,y)$ 表示为 $z=x+iy$。在这种情况下，实轴和虚轴分别是 $x$ 轴和 $y$ 轴。这个坐标平面就是复平面。根据距离公式，在复平面中，$|z|$ 可以看作是 $z$ 到原点 $o$ 的距离。此外，$z$ 到 $w$ 的距离是 $|z-w|$。加法在复平面上可以看作是 ${\mathbb{R}}^2$ 上的向量空间加法。此外，乘法的可视化方式更为复杂。因为极坐标可以表示 ${\mathbb{R}}^2$ 中的每一个点。对于任意 $z\in \mathbb{C}(z\ne 0)$，$|\frac{z}{|z|}|=|z|\cdot |\frac{1}{|z|}|=|z|\cdot \frac{1}{|z|}=1$。因此，存在 $\theta \in \mathbb{R}$，使得 $\frac{z}{|z|}=\cos \theta +i\sin \theta$。记 $e^{i\theta }=\cos \theta +i\sin \theta$。方程 $z=|z|e^{i\theta }$ 称为 $z\in \mathbb{C}$ 的极坐标形式。$z$ 的辐角是指 $\theta \in \mathbb{R}$ 这个角，记为 $\arg z$。利用三角恒等式可以验证 $e^{i{\theta }_1}{e}^{i{\theta }_2}=e^{i({\theta }_1+{\theta }_2)}$。因此，对于 $z,w\in \mathbb{C}$，$zw=|z|e^{i\arg z}\cdot |w|e^{i\arg w}=|zw|\cdot e^{i(\arg z+\arg w)}$。因此，$|w|$ 对 $z$ 进行缩放，$\arg w$ 使 $z$ 逆时针旋转从而得到 $zw$。此外，逆时针旋转 $(i=\frac{\pi }{2})$ 对应于 $z\to iz$。最后，关于实轴的反射对应于复共轭 $\overline{x+iy}=x-iy$。由于在复平面中距离可以用绝对值表示，三角不等式可以表示为 $|a+b|\le |a|+|b|$。从反向三角不等式的角度看，它可以写成 $-||a|-|b||\le |a-b|$，并且还可以推出 $|a|=|a-b+b|\le |a-b|+|b|$。由这些不等式可以得到 $|a|-|b|\le |a-b|$，$|b|=|b-a+a|\le |b-a|+|a|$，$-(|a|-|b|)=|b|-|a|\le |b-a|=|a-b|$。最重要的是，有 $|\text{Im}(a)|\le |a|$ 和 $|\text{Re}(a)|\le |a|$。收敛通常定义为：对于数列 $(W_n{)}_{n\in \mathbb{N}}\subset \mathbb{C}$，如果 ${\lim }_{n\to \infty }(W_n-Z)=0$，则称该数列收敛于 $Z\in \mathbb{C}$，记为 ${\lim }_{n\to \infty }{W}_n=Z$，其中 $z$ 称为该数列的极限。简而言之，可以写成对于 ${\forall }_n\ge n$，$(W_n{)}_{n\in \mathbb{N}}$ 收敛于 $Z$，$|W_n-Z|<C$。由于 $|\text{Im}(a)|\le |a|$，$|\text{Re}(a)|\le |a|$ 且 $|a|=(\text{Re}(a{)}^2+\text{Im}(a{)}^2{)}^{\frac{1}{2}}$，对于 ${\lim }_{n\to \infty }{W}_n=Z$，有 ${\lim }_{n\to \infty }\text{Re}(W_n)={\lim }_{n\to \infty }\text{Im}(W_n)=Z$。所以，复数的收敛概念与 ${\mathbb{R}}^2$ 上的收敛概念是一致的。$W_n\in \mathbb{C}$，当满足 ${\forall }_{\epsilon }>0$ 且 ${\exists }_N\in \mathbb{N}$，使得对于 ${\forall }_{n,m}\ge N$ 都有 $|W_n-W_m|<\epsilon$ 时，称 $(W_n)$ 为柯西序列。因此，柯西序列中的项最终会如人们所预期的那样。对于 $P_0\in \mathbb{C}$ 且 $r>0$，其中 $r$ 是以 $P_0$ 为圆心的开圆盘的半径，该圆盘可记为 P r ( P 0 ) : = { P ∈ C : ∣ P − P 0 ∣ < r } P_{r}(P_{0}):=\{P \in \mathbb {C}:|P-P_{0}|<r\} Pr​(P0​):={P∈C:∣P−P0​∣<r}。那么，这样的闭圆盘可以写成： P r ‾ ( P 0 ) : = { P ∈ C : ∣ P − P 0 ∣ ≤ r } \overline {P_{r}}(P_{0}):=\{P \in \mathbb {C}:|P-P_{0}|\leq r\} Pr​​(P0​):={P∈C:∣P−P0​∣≤r}。半径为 $r$ 的圆记为：$C_r(P_0)$，表示半径 $r$ 和圆心 $P_0$。然后我们可以只保留下面这部分作为记号。这里有一种简便方法来判断是否结束。复数序列的收敛是针对单位圆盘而言的： D : = D 1 ( 0 ) = { P ∈ C : ∣ P ∣ < 1 } D:=D_{1}(0)=\{P \in \mathbb {C}:|P|<1\} D:=D1​(0)={P∈C:∣P∣<1}。接下来，关注 $\mathbb{C}$ 的拓扑性质。由于复平面可以与 ${\mathbb{R}}^2$ 等同，并且具有相同的距离和收敛概念，这些性质只是从 ${\mathbb{R}}^2$ 到 $\mathbb{C}$ 的一种转换。最重要的是，从上面的例子可以看出，这不是简单的开集和闭集的二分法：整个集合可能是开集、闭集，或者既是开集又是闭集。通常，可以直接根据定义判断一个集合是否为开集。 { P ∈ C : ∣ P − P 0 ∣ < r } \{P \in \mathbb {C}:|P-P_{0}|<r\} {P∈C:∣P−P0​∣<r}。当上述任何内容对讨论不重要时，人们可以像定义实数一样来定义它。数列 { Z n } \{Z_{n}\} {Zn​} 收敛于 $\omega$ 的充要条件是其实部和虚部分别收敛于 $\omega$ 的实部和虚部。对于所有柯西序列 { Z n } \{Z_{n}\} {Zn​}，当 $n,m\to +\infty$ 时，有 $|Z_n-Z_m|\to 0$。实数集是完备的，实数的柯西序列收敛，而复数范围内的收敛等价于两个实数柯西序列的收敛。因此，复数集是完备的。

2. History of Complex numbers

复数的历史

The question of whether complex numbers exist is a tricky one. About three hundred years ago, the answer to this question was finally obtained. In fact, these numbers are quantities that arise in abundance in the study of mathematical disciplines, even though they may have meaning in reality or not at all.

复数是否存在是一个棘手的问题。大约三百年前，这个问题终于有了答案。事实上，这些数是在数学学科研究中大量出现的量，尽管它们在现实中可能有意义，也可能毫无意义。

Mathematics flourished gradually over thousands of years, but few people considered the limitations that only positive numbers and the real numbers imposed on mathematics and computation. In sixteenth century, imaginary number concepts were introduced. In the beginning, mathematicians did not even have the concept of negative numbers, let alone imaginary numbers. This led to the fact that even though they sometimes came up with similar ideas about complex numbers, they could not really touch on this concept.

数学在数千年间逐渐蓬勃发展，但很少有人思考仅使用正数和实数给数学及计算带来的局限。16 世纪，虚数概念被引入。起初，数学家们甚至没有负数的概念，更不用说虚数了。这就导致，即便他们有时会产生与复数类似的想法，却无法真正触及这一概念。

Nevertheless, certain “impossible” quantities were noticed by mathematicians working on cubic equations solutions. However, this almost completely foreign concept led them to believe that they had discovered something else. It is easy to see that the problem of “language barriers” permeates the work of mathematicians. As a result, it’s not rare for them to clash with each other over the years. There is no standard by which to judge how these mathematicians think, and this makes each mathematician start almost from scratch. These factors have led to slow progress on some complex theories.

尽管如此，研究三次方程解法的数学家们注意到了某些 “不可能” 的量。然而，这个几乎完全陌生的概念让他们以为自己发现了别的东西。很明显，“语言障碍” 问题贯穿于数学家们的工作中。因此，多年来他们之间发生分歧并不罕见。没有评判这些数学家思维方式的标准，这使得每位数学家几乎都要从头开始。这些因素导致一些复杂理论的发展十分缓慢。

Over four centuries, whatever the original purpose, with increasing and intensive research, this complex system has been widely accepted as a mathematical truth. The questioning and testing of mathematics by mathematicians has led to the further development of its proofs. The emergence of more abstract proofs has given mathematicians confidence and continued the development of complex systems. Actually, this system is called “complex analysis”. The development of complex numbers as an extension of rigorous proofs has provided new and expanding perspectives in many areas of mathematics [1-19].

在四个多世纪的时间里，不管最初的目的是什么，随着研究的不断深入，这个复数体系已被广泛接受为数学真理。数学家们对数学的质疑和检验推动了证明方法的进一步发展。更抽象的证明方法的出现让数学家们充满信心，也推动了复数体系的持续发展。实际上，这个体系被称为 “复分析”。复数作为严格证明的延伸，其发展为数学的许多领域提供了新的、更广阔的视角 [1-19]。

The first traces of imaginary numbers were found in Italy, hidden in the cubic equation. In about 1530s, Nicolo Tartaglia, the mathematical genius, came on the scene [1-19]. Tartaglia often called a “stammerer” because of his speech impediment and the massacre that took place in his hometown as a child. It ought to go without saying that Tartaglia was not taken seriously and his broad mathematical ideas were ignored.

虚数的最初踪迹在意大利被发现，隐藏在三次方程中。大约在 16 世纪 30 年代，数学天才尼科洛・塔尔塔利亚（Nicolo Tartaglia）登上了历史舞台 [1-19]。塔尔塔利亚常被称为 “口吃者”，这是因为他有语言障碍，且童年时家乡发生过屠杀事件。不言而喻，塔尔塔利亚没有受到重视，他广博的数学思想也被忽视了。

The use of method for cubic equations of a certain form was one of his primary achievements [1-19]. His technique in his era was revolutionary. Tartar created multiple equations to express those ideas. Firstly, check the depressed cubic equation given by
运用特定形式三次方程的解法是他的主要成就之一 [1-19]。在他那个时代，他的方法具有革命性。塔尔塔利亚创建了多个方程来表达这些想法。首先，来看一下缺二次项的三次方程 $$x^3+cx=d$$

Targaglia then defines two numbers whose difference is equal to d, and $(\frac{c}{3}{)}^3$ is their product. Let’s consider them as u and v. They become $u-v=d$, $uv=(\frac{c}{3}{)}^3$. The goal is still to find two numbers according to their product and difference. Tartaglia tried find the number $u+v$, and plus $u-v$ to solve variables. The method he used was to square the difference of u and v, add a quadruple product and then extract the square root [2]. Tartaglia found the solution creatively $x=\sqrt[3]{u}-\sqrt[3]{v}$.

然后塔尔塔利亚定义了两个数，它们的差等于 $d$，乘积为 $(\frac{c}{3}{)}^3$ 。我们把这两个数记为 $u$ 和 $v$ ，即 $u-v=d$，$uv=(\frac{c}{3}{)}^3$。目标仍是根据两个数的乘积和差来求出这两个数。塔尔塔利亚试图先求出 $u+v$，再结合 $u-v$ 来求解变量。他采用的方法是先对 $u$ 和 $v$ 的差进行平方，加上四倍的乘积，然后再开方 [2]。塔尔塔利亚创造性地得出了解 $x=\sqrt[3]{u}-\sqrt[3]{v}$。

To apply Tartaglia’s method on a numerical equation. Begin the equation like Equation 1, such as $x^3+3x=4$. In order to find x, you have to solve u and v, they have the difference 4 and product 1. According to Tartaglia’s method, solve $u+v$ firstly follow the step above. $u+v=\sqrt{(u-v{)}^2+4uv}=\sqrt{16+4}=\sqrt{20}=2\sqrt{5}$. As $u+v=2\sqrt{5}$ and $u-v=4$, u can be solved by those two equations by setting them equally using substitution, then find v with the same process. Therefore, the solution of this cubic function can be found $x=\sqrt[3]{\sqrt{5}+2}-\sqrt[3]{\sqrt{5}-2}$ .

要将塔尔塔利亚的方法应用于一个数字方程。以方程 $(1)$ 的形式为例，如 $x^3+3x=4$。为了求出 $x$，需要求解 $u$ 和 $v$，它们的差为 $4$，乘积为 $1$。根据塔尔塔利亚的方法，按照上述步骤先求出 $u+v$ 。$u+v=\sqrt{(u-v{)}^2+4uv}=\sqrt{16+4}=\sqrt{20}=2\sqrt{5}$。由于 $u+v=2\sqrt{5}$ 且 $u-v=4$，可以通过代入法联立这两个方程求解 $u$，再用同样的方法求出 $v$。因此，这个三次函数的解为 $x=\sqrt[3]{\sqrt{5}+2}-\sqrt[3]{\sqrt{5}-2}$。

In 1539, Tartaglia’s works were secretly passed on to Girolamo Cardano. However, Cardano published the method in his magnum opus Ars Magna [3]. As a result, after the publication of Ars Magna, the memory of Tartaglia’s method disappeared, and the technique has since been known forever as Cardano’s formula for depressed cubic. Cirolamo Cardano is considered as the most disgraceful mathematician. “Scoundrel” became synonymous with him and not without reason. In his magnum opus Ars Magna, the formula he developed for solving the depressed cubic using the Tartaglia’s method. Cardano nearly insulted the Italian mathematician Ferro when his book published. Nevertheless, the true creator Tartaglia was not praised.

1539 年，塔尔塔利亚的成果被偷偷传给了吉罗拉莫・卡尔达诺（Girolamo Cardano）。然而，卡尔达诺在他的巨著《大术》[3] 中发表了这个方法。结果，《大术》出版后，塔尔塔利亚的方法被人遗忘，从那以后，这个技巧就一直被称为卡尔达诺求解缺二次项三次方程的公式。吉罗拉莫・卡尔达诺被认为是最不光彩的数学家。“无赖” 成了他的代名词，这并非毫无缘由。在他的《大术》中，他用塔尔塔利亚的方法推导出求解缺二次项三次方程的公式。卡尔达诺的书出版时，他几乎冒犯了意大利数学家费罗（Ferro）。然而，真正的创造者塔尔塔利亚却没有得到赞誉。

Cardano’s technique is expressed in special form. He showed his method of finding z in terms of a real cube. The representation of his idea is shown in three-dimensional. Cardano’s approach is in the form of “rule” which is more generalized than using equations and steps. In his rule, he defined $x=t-u$ which is related to his cube, though Tartaglia’s solution for z is still used. After Cardano geometrically solving his cube, he came up with a general solution [1-19].

卡尔达诺的技巧以特殊形式呈现。他展示了通过一个实立方体来求 $z$ 的方法。他的想法以三维形式呈现。卡尔达诺的方法以 “规则” 形式呈现，比使用方程和步骤更具普遍性。在他的规则中，他定义 $x=t-u$，这与他的立方体相关，尽管仍使用塔尔塔利亚对 $z$ 的解法。卡尔达诺通过几何方法求解他的立方体后，得出了一个通用解法 [1-19]。

Roots calculated by Cardano’s method are shown in radical form, which it might cause the missing of some values. For the equation, $x=\sqrt[3]{\sqrt{5+2}}-\sqrt[3]{\sqrt{5}-2}$ mentioned formerly, it could be simplified to just $x=1$, however, by using Cardano’s method, it would not be shown. Cardano’s method is somehow not foolproof. In some cases, like what is in Equation 1, if containing some negativities, it would present results with radicand being negative without validating them. Since the non-real value are computed by the ‘rules of ordinary arithmetic’, scientists have been trying to prove their existence for more than three hundred years until they finally realized what they are proving is only imaginary quantity [1-19].

用卡尔达诺方法计算出的根以根式形式呈现，这可能会导致一些值的遗漏。对于前面提到的方程 $x=\sqrt[3]{\sqrt{5+2}}-\sqrt[3]{\sqrt{5}-2}$，它可以简化为 $x=1$，然而，用卡尔达诺的方法却无法得出这个结果。卡尔达诺的方法并非万无一失。在某些情况下，比如方程 $(1)$，如果包含一些负数，它会得出被开方数为负的结果，且没有对其进行验证。由于这些非实数值是按照 “普通算术规则” 计算出来的，科学家们花了三百多年试图证明它们的存在，最终才意识到他们所证明的只是虚数 [1-19]。

At that point, Cardano is not only trying to solve cubic equations in Tartaglia’s version but also in a new form of parameterizing as $a{x}^3+b{x}^2+cx+d=0$. However, by using his own method, Cardano continuously got roots that contains negative radicands, that he remained to assume no solution existing, while also missed the real roots that satisfies. His failure in solving cubic equations with imaginary quantities did not ruin his reputation and has also helped to sketch the blueprint of mathematical proof in the mid-sixteenth century. Cardano based his work on the structure of demonstration, rule, demonstration, rule, and so on (Ars Magna), and has helped to turn mathematics at that time from faith to proof. Using concrete proving, imaginary quantities could then be introduced.

那时，卡尔达诺不仅尝试用塔尔塔利亚的方法求解三次方程，还研究参数化为 $a{x}^3+b{x}^2+cx+d=0$ 这种新形式的三次方程。然而，用他自己的方法，卡尔达诺不断得到含有负被开方数的根，他一直认为不存在解，同时也错过了满足方程的实根。他在求解含虚数量的三次方程上的失败并没有毁掉他的声誉，反而为 16 世纪中期的数学证明勾勒出了蓝图。卡尔达诺的工作以论证、规则、论证、规则等结构为基础（《大术》），推动了当时的数学从依靠信念转向依靠证明。通过具体的证明，虚数得以引入。

Rafael Bombelli, the ‘last great sixteenth century Bolognese mathematician’, also an engineer and architect, with his L’Algebra, known as the turning point of the development in complex number, started the rigorous proving in imaginary quantities. Bombelli found that Cardano’s work useful but not necessarily correct as it is in ‘real’. As the founder of complexity, he decided to make up symbols himself [1-19]. He defined $\sqrt{-1}$ as pdm, in short of piu di meno in Italian, which means ‘plus of minus’ in English. Similarly, $-\sqrt{-1}$ was defined as mdm, meno di meno: ‘minus of minus’. At that period of time, things in mathematics are still describe in words instead of symbols as symbols were not fully circulated. In his language of mathematics, ‘plus of minus’ times ‘plus of minus’ equals to ‘minus’ where ‘minus’ is negative one.

拉斐尔・邦贝利（Rafael Bombelli），这位 “16 世纪博洛尼亚最后一位伟大的数学家”，同时也是一名工程师和建筑师，他的著作《代数学》被视为复数发展的转折点，开启了对虚数量的严格证明。邦贝利发现卡尔达诺的工作有一定价值，但从 “实数” 的角度看并不一定正确。作为复数领域的开创者，他决定自己创造符号 [1-19]。他把 $\sqrt{-1}$ 定义为 pdm，是意大利语 piu di meno 的缩写，在英语中意为 “加负”。类似地，$-\sqrt{-1}$ 被定义为 mdm，即 meno di meno，意为 “减负” 。在那个时期，数学中的内容仍常用文字描述，因为符号还未完全普及。在他的数学语言里，“加负” 乘以 “加负” 等于 “负”，这里的 “负” 就是负一。

After that, Bombelli starts to figure out cubic functions, especially those which can lead to roots of complex numbers. His procedure, accepting negative numbers as radicands which he named them as ‘linked radicals of a new type’, has well started the field of imaginary quantity as well as complex number. Through his own process of doing the irreducible cases of the cubic, he has shown the restriction of using Cardano’s method of finding roots [1-19]. By showing that the combination of complex numbers can lead to real quantities, he has also provided the missing steps in Cardano’s method.

此后，邦贝利开始研究三次函数，尤其是那些会得出复数根的函数。他的方法接受负数作为被开方数，并将其称为 “新型连根式”，这为虚数量和复数领域的发展奠定了良好基础。通过处理三次方程不可约情形的过程，他揭示了卡尔达诺求根方法的局限性 [1-19]。他证明了复数的组合可以得到实数，这也弥补了卡尔达诺方法中缺失的步骤。

Bombelli, after this series of things, has started his own study in mathematics which was basically continuing to prove that by Cardano’s recipe, real quantities can be manipulated, and that is what was known as conjugated complex number at present. In his belief, roots of cubic can be presented in form of $a+b\sqrt{-1}$, $a-b\sqrt{-1}$ where $a$ and $b$ are both real quantities. $2+\sqrt{-121}=(a+b\sqrt{-1}{)}^3=a(a^2-3b^2)+b(3a^2-b^2)\sqrt{-1}$. Bombelli through both sides cubic equation can be solved by $a$ and $b$, Since the integer values of $a$ and $b$ need to be considered, and $a$ must be either $1$ or $2$, meanwhile $b$ must be either $1$ or $11$, because both $2$ and $11$ are prime numbers. The only two combinations of solutions that satisfy these two equalities are $a=2$, $b=1$. Find the solutions for $a$ and $b$ by substituting them into the original formula. $x=(2+\sqrt{-1})+(2-\sqrt{-1})=4$ and $x=4$ just fits the previous inference of the original formula. This is why Bombelli was so great when he combined his theory with those of his predecessors to prove that real numbers are complex numbers. In fact, he demonstrated that any real number can be expressed as a complex number. This view was confirmed and accepted because he had the audacity to demonstrate a more specific proof idea. Instead of making rules out of calculations, he actually took something and showed why there was a misunderstanding about how the solutions of equations could be expressed differently. Thus, his introduction of complex numbers given mathematicians with a chance to extend cubic functions and to study algebra from a new perspective.

在经历了这一系列事情后，邦贝利开始了自己的数学研究，其核心是继续证明利用卡尔达诺的方法，实数可以进行相关运算，而这也就是如今所说的共轭复数。他认为，三次方程的根可以表示为 $a+b\sqrt{-1}$、$a-b\sqrt{-1}$ 的形式，其中 $a$ 和 $b$ 均为实数。对于 $2+\sqrt{-121}=(a+b\sqrt{-1}{)}^3=a(a^2-3b^2)+b(3a^2-b^2)\sqrt{-1}$，邦贝利通过这个等式两边来求解 $a$ 和 $b$ 。由于要考虑 $a$ 和 $b$ 的整数值，且 $a$ 只能是 $1$ 或 $2$，$b$ 只能是 $1$ 或 $11$，因为 $2$ 和 $11$ 都是质数。满足这两个等式的解只有两组，即 $a=2$，$b=1$。将其代入原公式可求出 $a$ 和 $b$ 的值。$x=(2+\sqrt{-1})+(2-\sqrt{-1})=4$，$x=4$ 恰好符合原公式之前的推导。这就是为什么邦贝利如此伟大，他将自己的理论与前辈们的理论相结合，证明了实数是复数的一种特殊情况。事实上，他证明了任何实数都可以表示为复数。这一观点得到了证实和认可，因为他大胆地展示了更具体的证明思路。他没有从计算中总结规则，而是实实在在地说明了为什么在方程解的不同表达方式上会存在误解。因此，他对复数的引入让数学家们有机会拓展三次函数，并从新的视角研究代数学。

By the end of the 17th century, Descartes was working on his symbolic rules, and also proposing the term “imaginable” for the complex, but without any positive discussion of the concept. In the 18th century, the concept of complex numbers came closer to our modern mathematical definition of them, they were more widely studied and adopted by popular mathematics, and during this century mathematicians had done enough to gain some confidence. The German mathematician Leibniz was one of the pioneers who took a step forward in the combination of numbers and shapes, especially in the geometric meaning of the conjugate of complex numbers. Inspired by Bombelli, Leibniz proposed that these cubic polynomials have either three real roots or two imaginary roots and one real root when studying the solutions of cubic polynomials. However, Leibniz’s real contribution lies in the following. The word “imaginable”. Unlike Bombelli, who coined the word out of necessity, Leibniz can be explicit the reason why these quantities are called imaginary numbers. Although he used these numbers, he could not confirm their nature value. This goes back to early mathematics built on geometry. So, since there was little support for the geometric meaning of complex numbers, he said they weren’t really numbers at all. It was from this perspective that the term “Imaginary” emerged and expanded its use.

到 17 世纪末，笛卡尔致力于研究符号法则，还提出用 “可想象的” 来形容复数，但并未对这一概念展开正面探讨。18 世纪，复数的概念更接近现代数学对其的定义，在大众数学领域得到了更广泛的研究和应用。在这一世纪里，数学家们做了大量工作，也因此获得了更多信心。德国数学家莱布尼茨是在数与形结合方面取得进展的先驱之一，尤其在复数共轭的几何意义研究上。受邦贝利启发，莱布尼茨在研究三次多项式的解时提出，这些三次多项式要么有三个实根，要么有两个虚根和一个实根。然而，莱布尼茨真正的贡献在于 “可想象的” 这个词。与因需要而创造词汇的邦贝利不同，莱布尼茨明确说明了这些量被称为虚数的原因。尽管他使用了这些数，但无法确定它们的实际价值。这要追溯到早期以几何为基础的数学。由于当时复数的几何意义缺乏支持，他认为这些数根本算不上真正的数。正是从这个角度出发，“虚数（Imaginary）” 这个术语出现并被广泛使用。

What makes Euler perhaps one of the greatest mathematical geniuses is that his ideas can be found in almost every field of mathematics today. He may not be known for his research on complex numbers, but he provided a basic but fundamental term for complex systems. He set $i=\sqrt{-1}$, and suggested that the best evidence that the problem could not be solved was to use this quantity to solve it. His creation of the word “i” did, however, make it simpler, but that did not prevent him from being one of the many mathematicians who still believed that complex numbers did not exist. As mathematicians began to study complex numbers in the 18th century, many reasonably came to believe that complex numbers had different “orders” or “types”, because there were many different classes of real numbers such as rational and irrational, natural and integer numbers. However, D’Alembert showed that there were many kinds of complex numbers, but they could all be expressed as $a+bi$, and this was the first time that a complex number was represented as two separate parts. This is also the contradiction of the complex number, which is infinite like the real number, but each of its elements can only be expressed in this single way [1-19].

欧拉堪称最伟大的数学天才之一，如今几乎在数学的各个领域都能看到他的思想。他或许并非因复数研究而闻名，但他为复数体系提供了一个基础且关键的符号。他设定 $i=\sqrt{-1}$，并提出用这个量来解决问题，恰恰是问题无法用常规方法解决的最好证明。他创造的 “$i$” 确实让相关表达更简便了，但这也不妨碍他和许多数学家一样，仍然认为复数并不存在。18 世纪，随着数学家们开始研究复数，许多人理所当然地认为复数有不同的 “阶” 或 “类型”，因为实数有很多不同类别，比如有理数和无理数、自然数和整数。然而，达朗贝尔指出，复数种类繁多，但都可以表示为 $a+bi$ 的形式，这是复数首次被表示为两个独立部分。这也是复数的矛盾之处，它像实数一样有无穷多种，但每个复数却只能用这一种形式表示 [1-19]。

3. Summary

总结

Complex analysis is a branch of mathematical analysis that focuses on operations on complex numbers. The theory of functions of a complex variable is another name for it. Holomorphic functions are the main area of study in the subject of complex analysis. These functions may take negative values, have differentiable features, and are defined on the complex plane. The Cauchy integral formula, the Laurent series expansion, the residue theorem, and other ideas and techniques are some that are frequently employed in research. Algebraic geometry, fluid dynamics, quantum mechanics, and other related fields have employed complex analysis extensively over the years in mathematics, physics, and engineering. In the 16th century, two Italian mathematicians named Girolamo Cardano and Raphael Bombelli made the discovery of complex numbers while seeking to solve a specific algebra. In the 19th century, Cauchy and Riemann developed their work. This article begins by looking into the arithmetic properties of complex numbers and then make a thorough investigation of historical development of complex numbers. In the future, we will report more on topological properties of complex field and elegant properties of holomorphic functions.

复分析是数学分析的一个分支，主要研究复数的运算。它也被称作复变函数论。全纯函数是复分析的主要研究对象。这些函数可以取负值，具有可微性，且定义在复平面上。柯西积分公式、洛朗级数展开、留数定理以及其他一些概念和技术在研究中经常被用到。多年来，复分析在数学、物理和工程领域被广泛应用于代数几何、流体动力学、量子力学等相关领域。16 世纪，两位意大利数学家吉罗拉莫・卡尔达诺和拉斐尔・邦贝利在求解特定代数问题时发现了复数。19 世纪，柯西和黎曼进一步发展了相关理论。本文先探讨了复数的算术性质，接着深入研究了复数的历史发展。未来，我们将进一步报道复数域的拓扑性质以及全纯函数的优美性质。

References

[1] Lang, S., 1999. Complex analysis by Serge Lang. London: Springer-Verlag.

[2] Stein, M. and Shakarchi, R., 2003. Princeton lectures in analysis, volume II: Complex analysis. Princeton: Princeton university press.

[3] Brannan, D., Clunie, J. and Kirwan, W., 1973. On the coefficient problem for functions of bounded boundary rotation. Annales Academiae Scientiarum Fennicae Series A I Mathematica, 1973, pp.1-18.

[4] Barnard, R.W. (1990). Open problems and conjectures in complex analysis. In: Ruscheweyh, S., Saff, E.B., Salinas, L.C., Varga, R.S. (eds) Computational Methods and Function Theory. Lecture Notes in Mathematics, vol 1435. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0087893.

[5] Barnard, R., Jayatilake, U. and Solynin, A., 2015. Brannan’s conjecture and trigonometric sums. Proceedings of the American Mathematical Society, 143 (5), pp.2117-2128.

[6] Aharonov, D. and Friedland, S., 1973. On an inequality connected with the coefficient conjecture for functions of bounded boundary rotation. Annales Academiae Scientiarum Fennicae Series A I Mathematica, 1973, pp.1-14.

[7] Milcetich, J., 1989. On a coefficient conjecture of Brannan. Journal of Mathematical Analysis and Applications, 139 (2), pp.515-522.

[8] Roger W. Barnard, Kent Pearce & William Wheeler (1997) On a Coefficient Conjecture of Brannan, Complex Variables, Theory and Application: An International Journal, 33:1-4, 51-61, DOI: 10.1080/17476939708815011

[9] Udaya C. Jayatilake (2013) Brannan’s conjecture for initial coefficients, Complex Variables and Elliptic Equations, 58:5, 685-694, DOI: 10.1080/17476933.2011.605445.

[10] Szász, R. On Brannan’s Conjecture. Mediterr. J. Math. 17, 38 (2020).

[11] Liu Shenghua, Pan Jifu, Zheng Jiyun, complex change function [M]. Changchun: Jilin Education Press,1988.

[12] Zhong Yuquan, complex function Theory (second edition) [M]. Beijing: Higher Education Press,1988.

[13] Tan Xiaohong, Wu Shengjian complex change function concise tutorial [M]. Beijing: Peking University Press,2006.

[14] Jerrld E Maislen, Basic complex analysis [M]. Freeman W H and Company, 1973.

[15] A Simple Proof of the Fundamental Cauchy-Goursat Theorem, Eliakim Hastings Moore, American Mathematical Society.

[16] Complex variables and applications / James Ward Brown, Ruel V. Churchill.-9th ed. 1221 Avenue of the Americas, New York, NY 10020.

[17] Taylor & Laurent theorem, Chandan kumar Department of physics S N Sinha College Jehanabad Introduction.

[18] A Formal Proof of Cauchy’s Residue Theorem, Wenda Li and Lawrence C. Paulson Computer Laboratory, University of Cambridge.

[19] Peters, Christen. The Reality of the Complex: The Discovery and Development of Imaginary Numbers. The College Mathematics Journal

The reality of Complex numbers

复数的意义

Yuran Cheng¹, Meng Lyu², *, and Ziqi Zhang³

¹Xi 'an Chongshi Middle School, Xian, China

²The Village high School, Houston, United States

³The High School Affiliated to Shaanxi Normal University, Xian, China

*Corresponding author Meng_Lyu@s.thevillageschool.com

Abstract

摘要

What is a complex number? The question may be well solved today, but about 400 years ago it would have baffled many mathematicians. For example, how was it possible to define the square root of a negative number, which was an “impossible” quantity at the time, but they were all real quantities in mathematics. Complex analysis is now recognized as a fundamental part of mathematics because of its many applications in physics and engineering as well as its connections to other branches of mathematics. By solving some equations of more than one degree, complex numbers were produced. Most mathematicians focused on real analysis and physics applications until the nineteenth century. Cauchy was the first figure to make a considerable effort in a challenging examination. He calculated a number of complex integrals using his integral formula, establishing the foundation for defining the operations and characteristics of complex analysis. Gauss furthered the development of the complex number geometrical theory. The first part of this essay focuses on the history discovery of mathematical and geometrical aspects of complex numbers. The topological features of the complex plane are then carefully understood after that. The beautiful characteristics of complex functions and series are finally compiled based on the topology of the complex field.

什么是复数？如今这个问题或许很容易解答，但在大约 400 年前，它却难倒了众多数学家。例如，如何定义负数的平方根呢？在当时，这是一个 “不可能” 的量，然而在数学中它们又都是真实存在的量。如今，复分析因其在物理、工程领域的诸多应用以及与数学其他分支的联系，被公认为数学的基础组成部分。通过求解一些高次方程，复数应运而生。直到 19 世纪，大多数数学家都专注于实分析和物理应用。柯西是首位在这一具有挑战性的研究中付出巨大努力的人。他运用自己的积分公式计算了许多复积分，为定义复分析的运算和特性奠定了基础。高斯进一步推动了复数几何理论的发展。本文第一部分聚焦于复数在数学和几何方面的历史发现。随后深入理解复平面的拓扑特征。最后，基于复数域的拓扑结构，梳理出复函数和级数的美妙特性。

Keywords: Complex numbers, topology, complex function.

关键词：复数；拓扑；复函数

1. Introduction

引言

Mathematicians have been troubled by these “impossible” quantities for centuries, and in fact, due to the limitations of early mathematics, mathematicians rarely considered the limitations of positive or real numbers [1 - 19]. They do not have any prospects for the development of these undefined quantities. As a result, complex systems have not been extensively exploited for thousands of years. It was only later that mathematicians became convinced that they had discovered something new, leading to developments in the field of complex numbers, such as the Cartesian coordinate system and the roots of functions.

几个世纪以来，数学家们一直被这些 “不可能” 的量所困扰。事实上，由于早期数学的局限性，数学家们很少考虑正数或实数的局限性 [1 - 19]。他们对这些未定义的量的发展毫无头绪。因此，复数系统在数千年里都未得到广泛探索。直到后来，数学家们确信自己发现了新的东西，才推动了复数领域的发展，比如笛卡尔坐标系和函数的根。

In recent centuries, mathematicians have explored complex numbers more and more deeply. They have gradually solved one problem after another in the field of complex analysis by proving them step by step, and at the same time opened up new horizons for mathematical research.

近几个世纪以来，数学家们对复数的探索越来越深入。他们通过逐步证明，逐渐解决了复分析领域中的一个又一个问题，同时也为数学研究开辟了新的视野。

2. The history of differential equation-Imaginary number

微分方程的历史 —— 虚数

Imaginary numbers appeared in early Italy, hidden in a cubic equation, and algebra was well developed in the 16th century [1 - 19]. At that time, Nicolo Tartaglia, a mathematician, used his unique method to solve a cubic equation [1 - 19]. His method was to replace the method of taking values by reasoning. He created many equations to further express his hypothesis, such as the reduced cubic equation shown below.

虚数最早出现在意大利，隐藏在一个三次方程中。16 世纪，代数学得到了很好的发展 [1 - 19]。当时，数学家尼科洛・塔尔塔利亚用他独特的方法求解三次方程 [1 - 19]。他的方法是用推理取代取值的方法。他创建了许多方程来进一步表达自己的假设，如下所示的简化三次方程。

In order to better carry out the next step of reasoning, Tartaglia defined two numbers, namely $u$ and $V$, defined their difference as $D$, and defined the legislation that their product is one third of $C$, so that we can get:the product and the difference between the two numbers and find the two numbers have to be taken, and this need to solve the quadratic equation, but fortunately, the mathematicians before have thought of the solution, and they have worked it out. Tartaglia also tried to find the value of $u+v$ according to previous methods, and then solved the problem according to the value of $u-v$ in the expression. His method of finding $u+v$ was to first find the square of $U-V$, then $4$ times $UV$, and then find the square root of this quantity [1 - 21]. Tartaglia’s solution is:By Tartaglia’s method, the equation is similar. To solve this equation, the values of $u$ and $v$ need to be found at first. According to Tartaglia’s scheme,there areBy finding the values of $u$ plus $v$ and $u$ minus $v$, you can plug them into the equation and find $u$ and $v$, respectively. And then the cubic equation can be solved. In 1539, however, Tartaglia’s scheme was made known to the Italian mathematician Girolamo Cardano, who mentioned it in his work Ars Magna [1 - 21]. Unfortunately, due to the betrayal of Girolamo Cardano, Tartaglia’s method never appeared again. So Cardano, the mathematician, took a lot of heat, and he published this book in an insult to the Italian mathematician Ferro, who had also developed a form of reduced cubed. But Cardano did not praise Tartaglia, although he was inspired by Tartaglia [5], and through his writings the reader can discover a different form from Tartaglia. Cardano solved for $x$ from the perspective of a three - dimensional cube, and the picture below shows his idea in the cube.

为了更好地进行下一步推理，塔尔塔利亚定义了两个数，即 $u$ 和 $V$，定义它们的差为 $D$，并规定它们的乘积是 $C$ 的三分之一，这样我们可以得到：需要求出这两个数的乘积和差，进而求出这两个数，而这需要求解二次方程。幸运的是，之前的数学家已经想到了解法，并计算出来了。塔尔塔利亚还尝试根据之前的方法求出 $u+v$ 的值，然后根据表达式中 $u-v$ 的值来解决问题。他求 $u+v$ 的方法是先求出 $(u-v{)}^2$，再求出 $4uv$，然后求出这个量的平方根 [1 - 21]。塔尔塔利亚的解法是：按照塔尔塔利亚的方法，该方程与之类似。要解这个方程，首先需要求出 $u$ 和 $v$ 的值。根据塔尔塔利亚的方案，有通过求出 $u+v$ 和 $u-v$ 的值，将它们代入方程，就可以分别求出 $u$ 和 $v$。然后就可以解出三次方程。然而，1539 年，塔尔塔利亚的方案被意大利数学家吉罗拉莫・卡尔达诺知晓，卡尔达诺在他的著作《大术》中提到了这个方案 [1 - 21]。不幸的是，由于吉罗拉莫・卡尔达诺的背叛，塔尔塔利亚的方法再也没有出现过。因此，这位数学家卡尔达诺受到了很多指责，他出版这本书也是对同样提出了一种简化三次方程形式的意大利数学家费罗的一种侮辱。但是卡尔达诺并没有赞扬塔尔塔利亚，尽管他受到了塔尔塔利亚的启发 [5]，而且通过他的著作，读者可以发现与塔尔塔利亚不同的形式。卡尔达诺从三维立方体的角度求解 $x$，下图展示了他在立方体中的思路。

Cardano’s method is not a very rigorous form, unlike Tartaglia’s strict algebraic steps and system of equations. In his method, he still used the solution to Tartaglia’s equation 2, except that in order to describe his cube, he defined $x=t-u$, and after he had solved the geometry, he then devised a general solution. Cardano’s “recipe” nevertheless depicts the roots in radical form, concealing many of the roots’ true integer values. For example, in equation 3, he represents the solution in its radical form. Cardano’s method is far from perfect. When $c$ and $d$ are negative, the approach will include the square root of a negative number in some versions of Equation 1. Cardano did not test these, but he did apply “laws of regular arithmetic” to imaginary quantities and shown that, if such numbers existed, they might satisfy the cube. However, mathematicians soon discovered that this was very impossible because they only employed methods for real numbers. Cardano, on the other hand, did not simply publish a version of Tartaglia’s cube solution in the form of equation 1. He also attempted to solve a novel cube form.

卡尔达诺的方法不像塔尔塔利亚那样有严格的代数步骤和方程组，不是一种非常严谨的形式。在他的方法中，他仍然使用了塔尔塔利亚方程 2 的解法，只是为了描述他的立方体，他定义了 $x=t-u$，在解决了几何问题后，他设计出了一个通用解法。然而，卡尔达诺的 “方法” 以根式形式描述根，掩盖了许多根的真实整数值。例如，在方程 3 中，他以根式形式表示解。卡尔达诺的方法远非完美。当 $c$ 和 $d$ 为负数时，在方程 1 的某些版本中，这种方法会包含负数的平方根。卡尔达诺没有检验这些情况，但他确实将 “常规算术法则” 应用于虚数量，并表明，如果这样的数存在，它们可能满足立方体方程。然而，数学家们很快发现这是非常不可能的，因为他们只使用了针对实数的方法。另一方面，卡尔达诺并没有简单地以方程 1 的形式发表塔尔塔利亚的三次方程解法。他还试图求解一种新的立方体形式。

He not only struck an impasse because of his single solution path, but he also “missed” the genuine answer that satisfies the cube. As a result, his comments on a subject he didn’t completely comprehend sounded unproductive. Rafael Bombelli later provided the missing steps in his approach. Cardano’s failure to convey the entire mathematical theory and rationalize it as acceptable did not harm his reputation, but rather demonstrates the structure of mathematical proofs in the mid - 16th century. It is essentially geometric, and the words are not stated explicitly, therefore the demonstration does not provide a compelling foundation for the broader rules that follow. Cardano described nearly all of his mathematical explanations as “demonstrations.” He did it with a specific goal in mind: to produce mathematical demonstrations so that “reasoning may encourage conviction.” Bombelli’s Algebra and the Beginnings of Complex Number Theory. Rafael Bombelli was certainly a character - he has been called “last great six - teenth century Bolognese mathematician”.Bombelli embodies the “ordinary man” fairly well, owing to his lack of academic specialization in mathematics. Unlike Cardano, Bomberi was unable to justify a solution that did not totally reflect the cube. L’Algebra, his work, was a watershed moment in the evolution of complex numbers, since it became a really rigorous representation of imaginary numbers. Language for the Unknown by Bombelli Bombelli’s language of unidentified numbers Bombelli regarded the amounts appearing in Cardano’s technique as both “actual” and evidence of a newly discovered mathematical potential.not merely a partially invalid method Because he was effectively building “from the ground up,” he chose to give these quantities names. He deciphered $-1$ as $pdm$, which is an acronym of the Italian $piudimeno$, which translates as “plus of minus.” Similarly, he called its inverse $(--1)$ $menodimeno$, which translates as “minus of minus” and is abbreviated as $dm$. At the time, mathematics was mostly represented by words rather than symbols. As a result, expressing the mathematical expressions of the unknowns in Bomberi would be relatively simple.

他不仅因为单一的求解路径陷入了僵局，还 “错过” 了满足立方体方程的真正答案。因此，他对一个自己没有完全理解的问题的评论听起来毫无成果。拉斐尔・邦贝利后来补充了他方法中缺失的步骤。卡尔达诺未能完整传达数学理论并使其合理化，这并没有损害他的声誉，反而展示了 16 世纪中期数学证明的结构。它本质上是几何的，表述并不明确，因此这个证明并没有为后续更广泛的规则提供令人信服的基础。卡尔达诺几乎将他所有的数学解释都描述为 “证明”。他这样做有一个明确的目标：给出数学证明，以便 “推理可以激发信念”。邦贝利的《代数学》与复数理论的开端。拉斐尔・邦贝利无疑是个独特的人物，他被称为 “16 世纪博洛尼亚最后一位伟大的数学家”。邦贝利很好地体现了 “普通人” 的特点，因为他在数学领域没有学术专长。与卡尔达诺不同，邦贝利无法为一个不能完全反映立方体的解法进行辩护。他的著作《代数学》是复数发展史上的一个转折点，因为它真正严谨地表述了虚数。邦贝利对未知量的表述。邦贝利将卡尔达诺方法中出现的量既视为 “真实的”，又视为新发现的数学潜力的证据，而不仅仅是一种部分无效的方法。因为他实际上是 “从头开始” 构建，所以他选择给这些量命名。他将 $-1$ 解读为 $pdm$，这是意大利语 $piudimeno$ 的缩写，意思是 “负加”。同样，他把它的倒数 $(--1)$ 称为 $menodimeno$，意思是 “负减”，缩写为 $dm$。当时，数学主要用文字而不是符号来表示。因此，在邦贝利的著作中，表达未知量的数学表达式相对简单。

Bomberi would write this as “seven plus three times plus of minus.” Bomberi not only coined the word imaginary quantity, but there is evidence that he recognized some of its features and associated operations. Bomberi “attacked the irreducible case of the cubic, which…leads to the complex number’s cube root.” He invented his own approach for calculating the cube equation, one that boldly admits the existence of the square root of a negative number. Consider Bomberi’s work in the irreducible situation of cubes, or cubes with three real roots. He first demonstrated how Cardano’s formula does not allow one to find these roots, which is why Tartaglia and Cardano labeled it irreducible. However, he demonstrated how combining imaginary roots can result in a real number, which provides the “missing step.” Take a look at the cubic equation provided by.

邦贝利会把这个写成 “七加三次负加”。邦贝利不仅创造了 “虚数量” 这个词，而且有证据表明他认识到了虚数量的一些特征和相关运算。邦贝利 “攻克了三次方程的不可约情形，这…… 导致了复数的立方根”。他发明了自己计算三次方程的方法，大胆地承认了负数平方根的存在。考虑邦贝利在立方体不可约情形（即有三个实根的立方体）中的工作。他首先证明了卡尔达诺公式无法求出这些根，这就是为什么塔尔塔利亚和卡尔达诺将其标记为不可约的原因。然而，他展示了如何将虚根组合起来得到一个实数，这就提供了 “缺失的步骤”。看一下给出的三次方程。

The solution is obtained by using Cardan’s approach to this problem.The equation stated, however, contains three real roots, namely $4$, $-2$ + Bomberi agreed that only one root was produced where three roots should have been obtained, and not these three roots.

用卡尔达诺的方法求解这个问题得到了解。然而，该方程有三个实根，即 $4$，$-2$ 加 邦贝利认同应该得到三个根，但实际上只得到了一个根，而且不是这三个根。

3. The reality of complex numbers

复数的意义

More empirically, Bombelli demonstrated that any real number may be stated in complex form. Bombelli’s theory on this subject was a mathematical breakthrough that added significant depth to the development of set theory, involving the organizational definition of real - number sets and the intimate relationship between real - number sets and imaginary sets.Creating a New Mathematical Language for Imaginary Numbers The Use of “Imaginary” by Leibniz. Complex number were seen in a more contemporary manner in the 18th century. In addition to producing more contemporary representations of imaginary numbers and expressions, Leibniz’s study of complex numbers advanced our grasp of number - theoretic features such touch conjugate as well as algebraic qualities like quadratic and binomial. Leibniz discovered a few intriguing characteristics. Positive rational numbers are produced by complex numbers, particularly by linear combinations of complex conjugates. For example: Leibniz also studied cubics, and the conclusion of the study is that “they will either have three real roots or two imaginary roots and one real root.” Leibniz out to demonstrate that, contrary to Bombelli’s assertion, Cardano’s formula was in fact universally valid and did not require redefinition. His use of the word “imaginary” contributed the most to the complex’s growth. In fact, this was one of the first times the phrases used to describe these numbers were utilized. Leibniz actually had a good reason for calling the amounts fictitious. “The Divine Spirit finds a sublime expression in the miracle of analysis…, that gap between being and not - being that we name the imaginative root of negative oneness,” he explained. These numbers were utilized by Leibniz, but he was still unaware of their nature; this is related to the fact that the first mathematics was based on geometry and what can be measured. He concluded that complex numbers weren’t actually numbers because there wasn’t much evidence to support their geometrical validity. This viewpoint is where the word “imaginary” first appeared and gained popularity. Euler invented the sign $i$ to symbolize the square root of the negative identity, $i=\sqrt{-1}$. This is very significant for the future research on the imaginary number, and the concise representation is widely used. Numerous mathematicians began to think that there could be various “orders” or “types” of complex numbers during the 18th century as the study of complex numbers progressed. Since there are many detailed classifications of real numbers, many mathematicians are curious about the classification of imaginary numbers. But d’Alembert proved that every imaginary number can be expressed in the form of $a+bi$, d’Alembert. The paradox of the set of complex numbers is that while each of its components can only be stated in this one form, the set is infinite, just like the real numbers.

从经验角度来看，邦贝利证明了任何实数都可以用复数形式表示。邦贝利在这个问题上的理论是一项数学突破，为集合论的发展增添了重要深度，涉及实数集的组织定义以及实数集与虚数集之间的密切关系。为虚数创建一种新的数学语言。莱布尼茨对 “虚数” 的使用。18 世纪，人们对复数有了更现代的认识。除了给出虚数和表达式更现代的表示形式，莱布尼茨对复数的研究推进了我们对诸如共轭等数论特征以及二次和二项式等代数性质的理解。莱布尼茨发现了一些有趣的特征。复数，特别是共轭复数的线性组合，可以产生正有理数。例如：莱布尼茨还研究了三次方程，研究结论是 “它们要么有三个实根，要么有两个虚根和一个实根”。莱布尼茨试图证明，与邦贝利的断言相反，卡尔达诺公式实际上是普遍有效的，无需重新定义。他对 “虚数” 这个词的使用对复数的发展贡献最大。事实上，这是首次使用描述这些数的术语之一。莱布尼茨称这些量为虚构的是有充分理由的。他解释说：“神的精神在分析的奇迹中找到了崇高的表达…… 存在与非存在之间的鸿沟，我们称之为负一的想象根”。莱布尼茨使用了这些数，但他仍然不清楚它们的本质；这与最初的数学基于几何和可测量的事物这一事实有关。他得出结论，复数实际上不是数，因为没有太多证据支持它们在几何上的有效性。“虚数” 这个词就是从这个观点中首次出现并流行起来的。欧拉发明了符号 $i$ 来表示负一的平方根，即 $i=\sqrt{-1}$。这对未来虚数的研究非常重要，这种简洁的表示法被广泛使用。18 世纪，随着复数研究的进展，许多数学家开始认为可能存在各种 “阶” 或 “类型” 的复数。由于实数有许多详细的分类，许多数学家对虚数的分类感到好奇。但是达朗贝尔证明了每个虚数都可以表示为 $a+bi$ 的形式。复数集的悖论在于，虽然它的每个元素只能用这一种形式表示，但这个集合却和实数集一样是无限的。

4. Complex plane and its topology

复平面及其拓扑结构

If the imaginary part of $z$ is $0$, the number $z$ is real, and if the real part of $z$ is $0$, we say that $z$ is imaginary. $C$ admits operations of addition, multiplication and complex conjugation. The natural expansions of the corresponding operations on $R$ with the rule that $i^2=-1$ are addition and multiplication. Furthermore, addition and multiplication are associative and commutative, admit identities ($0$ and $1$, respectively), admit increments $-z$ with ($forz\ne 0$), and obey the distributive law.

如果 $z$ 的虚部为 $0$，那么数 $z$ 是实数；如果 $z$ 的实部为 $0$，我们就说 $z$ 是虚数。复数集 $C$ 允许加法、乘法和复共轭运算。根据 $i^2=-1$ 的规则，实数集 $R$ 上相应运算的自然扩展就是复数的加法和乘法。此外，加法和乘法满足结合律和交换律，分别有单位元 $0$ 和 $1$，对于 $z\ne 0$ 有加法逆元 $-z$，并且满足分配律。

$C$ should be represented as a coordinate plane $R^2$ with the coordinate pair $z=x+iy$ displayed $(x,y)$. In this situation, the $x$ - axis is referred to as the real axis, the $y$ - axis is referred to as the imaginary axis, and the coordinate plane is referred to as the complex plane. Recalling the distance formula for $R^2$, we see that $|z|$ is the distance from $z$ to $0$ in the complex plane. More generally, $|z-w|$ is the distance from $z$ to $w$ in the complex plane. It is worth noting that addition corresponds to vector space addition on $R^2$ and may thus be viewed in this manner. To illustrate multiplication, remember that every point in $R^2$ may be defined by polar coordinates and convert this from $C$. Any $z\in C$. Define the same notation and terminology for subsets $\Omega \in C$. For $z_0\in C$ and $r>0$. The open disc of radius $r$ centered at $z_0$, D r ( z 0 ) : = { z ∈ C : ∣ z − z 0 ∣ < r } D_r (z_0):=\{z\in C:\vert z - z_0\vert<r\} Dr​(z0​):={z∈C:∣z−z0​∣<r}. The closed disc of radius $r$ centered at $z_0$, D ‾ r ( z 0 ) : = { z ∈ C : ∣ z − z 0 ∣ ≤ r } \overline {D}_r (z_0):=\{z\in C:\vert z - z_0\vert\leq r\} Dr​(z0​):={z∈C:∣z−z0​∣≤r}. The circle disc of radius $r$ centered at $z_0$, C r ( z 0 ) : = { z ∈ C : ∣ z − z 0 ∣ = r } C_r (z_0):=\{z\in C:\vert z - z_0\vert = r\} Cr​(z0​):={z∈C:∣z−z0​∣=r}. The set of points in mathematics whose distance from a given point in the plane, $P$, is less than one is known as the open unit disk (or disc): D 1 ( P ) = { Q : ∣ P − Q ∣ < 1 } D_1 (P)=\{Q:\vert P - Q\vert<1\} D1​(P)={Q:∣P−Q∣<1}. The closed unit disk around $P$ is the set of points whose distance from $P$ is less than or equal to one: D ‾ 1 ( P ) = { Q : ∣ P − Q ∣ ≤ 1 } \overline {D}_1 (P)=\{Q:\vert P - Q\vert\leq1\} D1​(P)={Q:∣P−Q∣≤1}.

复数集 $C$ 应该表示为坐标平面 $R^2$，坐标对 $z=x+iy$ 表示为 $(x,y)$。在这种情况下，$x$ 轴被称为实轴，$y$ 轴被称为虚轴，坐标平面被称为复平面。回顾 $R^2$ 的距离公式，我们可以看到 $|z|$ 是复平面上 $z$ 到 $0$ 的距离。更一般地，$|z-w|$ 是复平面上 $z$ 到 $w$ 的距离。值得注意的是，加法对应于 $R^2$ 上的向量空间加法，因此可以从这个角度来看待。为了解释乘法，记住 $R^2$ 中的每个点都可以用极坐标定义，并从复数 $C$ 进行转换。对于任意 $z\in C$ 。对复数集 $C$ 的子集 $\Omega$ 定义相同的符号和术语。对于 $z_0\in C$ 和 $r>0$，以 $z_0$ 为圆心、半径为 $r$ 的开圆盘 D r ( z 0 ) : = { z ∈ C : ∣ z − z 0 ∣ < r } D_r (z_0):=\{z\in C:\vert z - z_0\vert<r\} Dr​(z0​):={z∈C:∣z−z0​∣<r}；以 $z_0$ 为圆心、半径为 $r$ 的闭圆盘 D ‾ r ( z 0 ) : = { z ∈ C : ∣ z − z 0 ∣ ≤ r } \overline {D}_r (z_0):=\{z\in C:\vert z - z_0\vert\leq r\} Dr​(z0​):={z∈C:∣z−z0​∣≤r}；以 $z_0$ 为圆心、半径为 $r$ 的圆 C r ( z 0 ) : = { z ∈ C : ∣ z − z 0 ∣ = r } C_r (z_0):=\{z\in C:\vert z - z_0\vert = r\} Cr​(z0​):={z∈C:∣z−z0​∣=r}。平面上到给定点 $P$ 的距离小于 $1$ 的点的集合称为开单位圆盘： D 1 ( P ) = { Q : ∣ P − Q ∣ < 1 } D_1 (P)=\{Q:\vert P - Q\vert<1\} D1​(P)={Q:∣P−Q∣<1}；点 $P$ 周围的闭单位圆盘是到 $P$ 的距离小于或等于 $1$ 的点的集合： D ‾ 1 ( P ) = { Q : ∣ P − Q ∣ ≤ 1 } \overline {D}_1 (P)=\{Q:\vert P - Q\vert\leq1\} D1​(P)={Q:∣P−Q∣≤1}。

The subject of topology focuses on certain characteristics of geometric spaces or figures that hold true despite repeated form changes. It ignores the sizes and forms of the items and only takes into account their spatial connections. In topology, the important topological properties include connectivity and compactness. It is simply a translation from $R^2$ to $C$.

拓扑学主要研究几何空间或图形的某些特性，这些特性在形状反复变化后仍然保持不变。它忽略物体的大小和形状，只考虑它们的空间关系。在拓扑学中，重要的拓扑性质包括连通性和紧致性。这只是从 $R^2$ 到复数集 $C$ 的一种转换。

In topology and related mathematical fields, topological properties or topological invariants are properties of topological spaces that are invariant under homeomorphisms. Or a topological property is a class of topological spaces that are closed under a homeomorphism. That is, if a space $X$ has this property, then all spaces that are homeomorphic to $X$ have this property, then the property of this space is a topological property. In layman’s terms, a topological property is a spatial property that can be expressed in terms of open sets.

在拓扑学及相关数学领域中，拓扑性质或拓扑不变量是拓扑空间在同胚变换下保持不变的性质。或者说，拓扑性质是一类在同胚变换下封闭的拓扑空间的性质。也就是说，如果空间 $X$ 具有这个性质，那么所有与 $X$ 同胚的空间都具有这个性质，那么这个空间的性质就是拓扑性质。通俗地说，拓扑性质是一种可以用开集来描述的空间性质。

So that, it can be defined $\Omega \in C$ is open if for all $z\in \Omega$ there exists $r>0$ such that ，we say $\Omega$ is closed if its complement ${\Omega }^c=C\setminus \Omega$ is open.

因此，可以定义：如果对于所有 $z\in \Omega$，都存在 $r>0$，使得 ，那么称 $\Omega \in C$ 是开集；如果它的补集 ${\Omega }^c=C\setminus \Omega$ 是开集，那么称 $\Omega$ 是闭集。

A common problem in topology is to determine whether two topological spaces are homeomorphic. To prove that two spaces are not homeomorphic, it is sufficient to find the topological property that they do not share.

拓扑学中的一个常见问题是确定两个拓扑空间是否同胚。要证明两个空间不同胚，只需找到它们不共有的拓扑性质即可。

A set $\Omega \in C$ is closed if for any convergent sequence $(z_n)\in \Omega$ we have ${\lim }_{n\to \infty }{z}_n\in \Omega$。

集合 $\Omega \in C$ 是闭集，当且仅当对于 $\Omega$ 中的任何收敛序列 $(z_n)$，都有 ${\lim }_{n\to \infty }{z}_n\in \Omega$。

Proof: Assume $\Omega$ is closed and let $(z_n)\in \Omega$ be a convergent sequence. If $w\notin \Omega$, then since ${\Omega }^c$ is open there exists $r>0$ so that $D_r(w)\subseteq {\Omega }^c$.

证明：假设 $\Omega$ 是闭集，$(z_n)\in \Omega$ 是一个收敛序列。如果 $w\notin \Omega$，那么由于 ${\Omega }^c$ 是开集，存在 $r>0$，使得 $D_r(w)\subseteq {\Omega }^c$。

5. Functions on the complex plane

复平面上的函数

When you study functions, there are certain properties that are very important, such as continuity and differentiability. The function $f$ is continuous at $z_0\in C$ if ${\lim }_{z\to z_0}f(z)=f(z_0)$ exists.

在研究函数时，有一些性质非常重要，比如连续性和可微性。如果 ${\lim }_{z\to z_0}f(z)=f(z_0)$ 存在，那么函数 $f$ 在 $z_0\in C$ 处连续。

More explicit: For any $\gamma >0$, there exists $\sigma =\sigma (\gamma )$ such that: $|f(z)-f(z_0)|<\gamma$ when $|z-z_0|<\sigma (\gamma )$. A continuous function $f$ defined on a compact set $\tau$, then it has its maximum and minimum value.

更明确地说：对于任意 $\gamma >0$，存在 $\sigma =\sigma (\gamma )$，使得当 $|z-z_0|<\sigma (\gamma )$ 时，$|f(z)-f(z_0)|<\gamma$。定义在紧集 $\tau$ 上的连续函数 $f$，有最大值和最小值。

The function $f$ is holomorphic at point $z_0\in C$ if ${\lim }_{z\to z_0}\frac{f(z)-f(z_0)}{z-z_0}$ exists and is denoted as $f^{\prime }(z_0)$。

如果 ${\lim }_{z\to z_0}\frac{f(z)-f(z_0)}{z-z_0}$ 存在，并记为 $f^{\prime }(z_0)$，那么函数 $f$ 在点 $z_0\in C$ 处是全纯的。

More explicit: For any $\gamma >0$ there exists $\sigma =\sigma (\gamma )$ such that: $\left|\frac{f(z)-f(z_0)}{z-z_0}-f^{\prime }(z_0)\right|<\gamma$ when $|z-z_0|<\sigma (\gamma )$. The function $f$ is said to be holomorphic if it is holomorphic on $C$. As for the holomorphic function $f$ and closed path $\delta$ in the complex plane, we have: ${\int }_{\delta }f=0$.When $m$, $n$ are holomorphic functions defined in set $\gamma$ it follows that: $m+n$ is holomorphic in $\gamma$ and $(m+n{)}^{\prime }=m^{\prime }+n^{\prime }$. $mn$ is holomorphic in $\gamma$ and $(mn{)}^{\prime }=m^{\prime }n+m{n}^{\prime }$; when $a\in \gamma$ and $n(a)\ne 0$, $\frac{m(a)}{n(a)}$ is holomorphic and ${\left(\frac{m(a)}{n(a)}\right)}^{\prime }=\frac{m^{\prime }(a)n(a)-n^{\prime }(a)m(a)}{n(a{)}^2}$.When function $m:a\to b$ and $n:b\to c$ is holomorphic, it holds that $(n\circ m{)}^{\prime }(z)=n^{\prime }(m(z))m^{\prime }(z)$ for all $z\in a$.

更明确地说：对于任意 $\gamma >0$，存在 $\sigma =\sigma (\gamma )$，使得当 $|z-z_0|<\sigma (\gamma )$ 时，$\left|\frac{f(z)-f(z_0)}{z-z_0}-f^{\prime }(z_0)\right|<\gamma$。若函数 $f$ 在 $C$ 上全纯，则称 $f$ 是全纯函数。对于复平面上的全纯函数 $f$ 和闭曲线 $\delta$，有 ${\int }_{\delta }f=0$ 。当 $m$、$n$ 是定义在集合 $\gamma$ 上的全纯函数时，$m+n$ 在 $\gamma$ 上是全纯的，且 $(m+n{)}^{\prime }=m^{\prime }+n^{\prime }$；$mn$ 在 $\gamma$ 上是全纯的，且 $(mn{)}^{\prime }=m^{\prime }n+m{n}^{\prime }$；当 $a\in \gamma$ 且 $n(a)\ne 0$ 时，$\frac{m(a)}{n(a)}$ 是全纯的，且 ${\left(\frac{m(a)}{n(a)}\right)}^{\prime }=\frac{m^{\prime }(a)n(a)-n^{\prime }(a)m(a)}{n(a{)}^2}$。当函数 $m:a\to b$ 和 $n:b\to c$ 是全纯函数时，对于所有 $z\in a$，有 $(n\circ m{)}^{\prime }(z)=n^{\prime }(m(z))m^{\prime }(z)$。

6. Summary

总结

A complex number is what? Today the problem may be easily resolved, but 400 years ago it would have stumped many mathematicians. How was it feasible, for instance, to define the square root of a negative integer, which was considered to be an “impossible” quantity at the time, even if all mathematical values were real? Because of its numerous uses in physics and engineering as well as its linkages to other areas of mathematics, complex analysis is now regarded as a fundamental component of mathematics. Complex numbers were created by resolving some equations with many degrees. Up to the eighteenth century, the majority of mathematicians concentrated on real analysis and physics applications. The first person to put in a significant effort in a difficult test was Cauchy. Using his integral formula, he computed a number of complex integrals, laying the groundwork for defining the functions and traits of complex analysis. The geometrical theory of complex numbers was advanced by Gauss. This essay’s first section focuses on the historical discovery of complex numbers’ mathematical and geometrical properties. The complex plane’s topological characteristics are then thoroughly understood. Finally, based on the topology of the complex field, the lovely properties of complex functions and series are assembled.

什么是复数？如今这个问题或许很容易解决，但在 400 年前，它会难倒许多数学家。例如，如何定义负数的平方根呢？在当时，这被认为是一个 “不可能” 的量，即便所有数学值都是实数。由于复数在物理和工程领域有众多应用，并且与数学的其他分支存在联系，复分析如今被视为数学的一个基本组成部分。复数是通过求解一些高次方程产生的。直到 18 世纪，大多数数学家都专注于实分析和物理应用。柯西是第一个在这项具有挑战性的研究中付出巨大努力的人。他利用自己的积分公式计算了许多复积分，为定义复分析的函数和特性奠定了基础。高斯推动了复数几何理论的进一步发展。本文的第一部分重点讲述了复数在数学和几何方面的历史发现。随后深入理解了复平面的拓扑特征。最后，基于复数域的拓扑结构，整理出了复函数和级数的美妙特性。

References

[1] Lang, S., 1999. Complex analysis by Serge Lang. London: Springer-Verlag.
[2] Stein, M. and Shakarchi, R., 2003. Princeton lectures in analysis, volume II: Complex analysis. Princeton: Princeton university press.
[3] Brannan, D., Clunie, J. and Kirwan, W., 1973. On the coefficient problem for functions of bounded boundary rotation. Annales Academiae Scientiarum Fennicae Series A I Mathematica, 1973, pp.1-18.
[4] Barnard, R.W. (1990). Open problems and conjectures in complex analysis. In: Ruscheweyh, S., Saff, E.B., Salinas, L.C., Varga, R.S. (eds) Computational Methods and Function Theory. Lecture Notes in Mathematics, vol 1435. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0087893.
[5] Barnard, R., Jayatilake, U. and Solynin, A., 2015. Brannan’s conjecture and trigonometric sums. Proceedings of the American Mathematical Society, 143(5), pp.2117-2128.
[6] Aharonov, D. and Friedland, S., 1973. On an inequality connected with the coefficient conjecture for functions of bounded boundary rotation. Annales Academiae Scientiarum Fennicae Series A I Mathematica, 1973, pp.1-14.
[7] Milcetich, J., 1989. On a coefficient conjecture of Brannan. Journal of Mathematical Analysis and Applications, 139(2), pp.515-522.
[8] Roger W. Barnard, Kent Pearce & William Wheeler (1997) On a Coefficient Conjecture of Brannan, Complex Variables, Theory and Application: An International Journal, 33:1-4, 51-61, DOI: 10.1080/17476939708815011
[9] Udaya C. Jayatilake (2013) Brannan’s conjecture for initial coefficients, Complex Variables and Elliptic Equations, 58:5, 685-694, DOI: 10.1080/17476933.2011.605445.
[10] Szász, R. On Brannan’s Conjecture. Mediterr. J. Math. 17, 38 (2020).
[11] Liu Shenghua, Pan Jifu, Zheng Jiyun, complex change function [M]. Changchun: Jilin Education Press,1988。
[12] Zhong Yuquan, complex function Theory (second edition) [M]. Beijing: Higher Education Press,1988.
[13] Tan Xiaohong, Wu Shengjian complex change function concise tutorial [M]. Beijing: Peking University Press,2006.
[14] Jerrld E Maislen, Basic complex analysis[M]. Freeman W H and Company, 1973.
[15] A Simple Proof of the Fundamental Cauchy-Goursat Theorem, Eliakim Hastings Moore, American Mathematical Society.
[16] Complex variables and applications / James Ward Brown, Ruel V. Churchill.—9th ed. 1221 Avenue of the Americas, New York, NY 10020.
[17] Taylor & Laurent theorem, Chandan kumar Department of physics S N Sinha College Jehanabad Introduction.
[18] A Formal Proof of Cauchy’s Residue Theorem, Wenda Li and Lawrence C. Paulson Computer Laboratory, University of Cambridge.

via:

• Complex number and its discovery history
  https://www.researchgate.net/publication/369457427_Complex_number_and_its_discovery_history

• The reality of Complex numbers
  https://www.researchgate.net/publication/369468392_The_reality_of_Complex_numbers

• 从三次方程到复平面：复数概念的奇妙演进（一）-CSDN博客
  https://blog.csdn.net/u013669912/article/details/147179204

• 从三次方程到复平面：复数概念的奇妙演进（二）-CSDN博客
  https://blog.csdn.net/u013669912/article/details/147194451

• 从三次方程到复平面：复数概念的奇妙演进（三）-CSDN博客
  https://blog.csdn.net/u013669912/article/details/147193352