Complex Functions: Key Points | 复变函数：考点精讲

📚 Complex Functions: Key Points | 复变函数：考点精讲

In KS3 mathematics, you are used to working with real numbers and simple functions. Now it is time to extend your understanding to complex numbers – numbers that include the imaginary unit i, where i² = –1. A complex function takes a complex number as its input and gives another complex number as the output. This advanced topic builds the bridge between real-world calculations and deeper mathematical ideas. Although it is an extension beyond the usual curriculum, grasping the basics of complex functions will sharpen your algebraic skills and open the door to fascinating patterns like the Mandelbrot set.

在KS3数学中，你已经熟悉了实数和简单的函数。现在我们要把理解拓展到复数——也就是包含虚数单位 i (i² = –1) 的数。复变函数以复数为输入，并输出另一个复数。这个进阶主题连接着现实世界的计算和更深层的数学思想。虽然它超出了常规课程的范围，但掌握复变函数的基础知识能磨砺你的代数技巧，并为你打开通往曼德勃罗集等迷人图案的大门。

1. Complex Numbers – The Building Blocks | 复数——构建基石

A complex number is written as z = a + bi, where a is the real part and b is the imaginary part. Here i is defined by i² = –1. Real numbers are just complex numbers with b = 0, and pure imaginary numbers have a = 0.

复数记作 z = a + bi，其中 a 是实部，b 是虚部。这里 i 满足 i² = –1。实数就是 b = 0 的复数，纯虚数则是 a = 0 的复数。

Addition and subtraction are straightforward: (a+bi) ± (c+di) = (a±c) + (b±d)i. Multiplication uses the distributive law and i² = –1: (a+bi)(c+di) = (ac–bd) + (ad+bc)i.

加减法很直接：(a+bi) ± (c+di) = (a±c) + (b±d)i。乘法使用分配律并利用 i² = –1: (a+bi)(c+di) = (ac–bd) + (ad+bc)i。

Conjugate: The complex conjugate of z = a+bi is z̄ = a–bi. The product z·z̄ = a² + b² is always a real number.
共轭复数：z = a+bi 的共轭为 z̄ = a–bi。乘积 z·z̄ = a² + b² 总是实数。

2. What Is a Complex Function? | 什么是复变函数？

A complex function f is a rule that assigns a complex output w = f(z) to every complex input z from a chosen set. You can think of it as a machine: put z = x+iy in, and get w = u+iv out, where u and v are real numbers that depend on x and y.

复变函数 f 是一种规则，它将每个取自某个集合的复数输入 z 对应到一个复数输出 w = f(z)。你可以把它想象成一台机器：放入 z = x+iy，得到 w = u+iv，其中 u 和 v 是依赖于 x 和 y 的实数。

For example, f(z) = z² means that if z = 1+2i, then f(z) = (1+2i)² = 1 + 4i + 4i² = 1 + 4i – 4 = –3 + 4i. So the output is –3 + 4i.

例如，f(z) = z² 意味着如果 z = 1+2i，那么 f(z) = (1+2i)² = 1 + 4i + 4i² = 1 + 4i – 4 = –3 + 4i。即输出为 –3 + 4i。

3. Real and Imaginary Parts of a Complex Function | 复变函数的实部与虚部

Every complex function f(z) can be split into two real-valued functions: f(x+iy) = u(x,y) + i v(x,y). The component u is the real part, and v is the imaginary part. Understanding these parts helps us graph and analyse the function.

每个复变函数 f(z) 可以拆分成两个实值函数：f(x+iy) = u(x,y) + i v(x,y)。分量 u 是实部，v 是虚部。理解这两个部分有助于我们绘制函数图像并进行分析。

For f(z) = z², expand (x+iy)² = x² – y² + i(2xy). Thus u(x,y) = x² – y², and v(x,y) = 2xy.

对于 f(z) = z²，展开 (x+iy)² = x² – y² + i(2xy)。因此 u(x,y) = x² – y²，v(x,y) = 2xy。

f(z)	u(x,y) Real Part 实部	v(x,y) Imaginary Part 虚部
z	x	y
z²	x² – y²	2xy
z³	x³ – 3xy²	3x²y – y³

4. Simple Examples: f(z)=z and f(z)=z+c | 简单例子：f(z)=z 与 f(z)=z+c

The identity function f(z) = z leaves every complex number unchanged. Its real and imaginary parts are simply u = x, v = y.

恒等函数 f(z) = z 对每个复数都不作改变。它的实部和虚部分别就是 u = x，v = y。

Adding a constant c = p+qi gives f(z) = z + c. Then u = x+p and v = y+q, which is a translation (shift) in the complex plane.

加上一个常数 c = p+qi 得到 f(z) = z + c。那么 u = x+p，v = y+q，这在复平面上表现为一次平移。

These simple functions preserve shapes: a square remains a square, just moved to a new location.

这些简单的函数保形：一个正方形还是正方形，只是被移动到了新位置。

5. Exploring f(z) = z² | 探索 f(z) = z²

The function f(z) = z² squares the magnitude and doubles the argument (angle). In polar form, if z = r(cosθ + i sinθ), then z² = r²(cos2θ + i sin2θ).

函数 f(z) = z² 会将模平方，并将辐角加倍。在极坐标形式中，若 z = r(cosθ + i sinθ)，则 z² = r²(cos2θ + i sin2θ)。

Geometrically, a straight line through the origin becomes a ray at twice the angle. A circle of radius R maps to a circle of radius R², but wrapped twice around.

从几何上看，通过原点的直线会变成一条角度加倍的射线。半径为 R 的圆映射为一个半径为 R² 的圆，但会绕原点两圈。

This doubling action leads to interesting symmetry: two opposite points z and –z both map to the same value z², so the function is not one-to-one.

这种加倍作用导致了有趣的对称性：两个相反的点 z 和 –z 都映射到同一个值 z²，因此函数不是一一映射。

6. The Reciprocal Function f(z) = 1/z | 倒数函数 f(z) = 1/z

For z ≠ 0, the function f(z) = 1/z is defined. Using the conjugate, we can write 1/z = z̄ / (z·z̄) = (x–iy)/(x²+y²). So u = x/(x²+y²) and v = –y/(x²+y²).

对于 z ≠ 0，定义函数 f(z) = 1/z。利用共轭复数，可写成 1/z = z̄ / (z·z̄) = (x–iy)/(x²+y²)。因此 u = x/(x²+y²)，v = –y/(x²+y²)。

This function swaps the interior and exterior of the unit circle. It turns circles not passing through the origin into other circles, and lines not through the origin into circles passing through the origin.

这个函数交换了单位圆的内部和外部。它将不通过原点的圆变成其他的圆，并将不通过原点的直线变成通过原点的圆。

It is an example of a Möbius transformation, a type of function that preserves angles but not sizes.

它属于莫比乌斯变换，这类函数保角但不保大小。

7. Graphing Complex Functions – A Challenge | 复变函数的图像——一大挑战

Since both input and output are 2‑dimensional (complex plane), a full graph would need 4 dimensions. We often visualise how regions or grids transform instead.

由于输入和输出都是二维的（复平面），要画出完整的图像需要四维空间。因此我们常转而可视化区域或网格如何变形。

One method is to draw a grid on the z‑plane and see its image on the w‑plane. For f(z)=z², a rectangular grid becomes a set of orthogonal parabolas.

一种方法是在 z 平面上画出网格，再观察它在 w 平面上的像。对于 f(z)=z²，矩形网格会变形成一组正交的抛物线。

Colour coding is another popular technique: each point in the output plane is coloured based on its input argument or magnitude.

颜色编码是另一种常用技巧：根据输入点的辐角或模长给输出平面上每个点上色。

8. Continuity of Complex Functions | 复变函数的连续性

A complex function is continuous at a point if small changes in input produce small changes in output. More precisely, f(z) is continuous at z₀ if for every ε > 0 there is a δ > 0 such that |z–z₀| < δ implies |f(z)–f(z₀)| < ε.

若输入发生微小变化，输出也只发生微小变化，则复变函数在该点连续。更精确地说，f(z) 在 z₀ 连续是指：对任意 ε > 0，存在 δ > 0，当 |z–z₀| < δ 时，有 |f(z)–f(z₀)| < ε。

Polynomials like f(z)=z², f(z)=z³+2z are continuous everywhere. The reciprocal function 1/z is continuous except at z=0.

多项式函数如 f(z)=z²、f(z)=z³+2z 处处连续。倒数函数 1/z 除 z=0 外皆连续。

Continuity is the first step toward differentiability – a key property in complex analysis called holomorphy.

连续性是通向可微性的第一步——这是复分析中称为全纯的关键性质。

9. Modulus and Argument Transformations | 模与辐角的变换

The modulus of a complex function |f(z)| tells us how much the function stretches lengths. For f(z)=z², |f(z)| = |z|². The argument gives the rotation: arg(f(z)) = 2 arg(z).

复变函数的模 |f(z)| 告诉我们函数对长度的拉伸程度。对于 f(z)=z²，|f(z)| = |z|²。辐角给出了旋转量：arg(f(z)) = 2 arg(z)。

For f(z)=1/z, |f(z)| = 1/|z|, and arg(1/z) = –arg(z). So the reciprocal reflects points in the unit circle and flips the angle sign.

对于 f(z)=1/z，|f(z)| = 1/|z|，且 arg(1/z) = –arg(z)。所以倒数函数将点关于单位圆反演并反转角度符号。

Understanding modulus and argument transforms helps predict the shape of the image without computing real and imaginary parts.

理解模和辐角的变换有助于预判像的形状，而无需计算实部和虚部。

10. Key Skills and Practice Tips | 关键技能与练习建议

To master the basics of complex functions, practise separating f(z) into u+iv form. Learn to express z in both Cartesian (x+iy) and polar forms.

要掌握复变函数的基础，请练习将 f(z) 分离成 u+iv 的形式。学会用笛卡儿形式 (x+iy) 和极坐标形式表示 z。

Work through squaring, cubing, and reciprocal functions step by step. Check that the modulus and argument follow the rules outlined above.

逐步演算平方、立方和倒数函数。检验模与辐角是否符合上述规则。

Simplify powers of i: i, i²=–1, i³=–i, i⁴=1, then repeats.
化简 i 的幂：i, i²=–1, i³=–i, i⁴=1，然后循环。
Use conjugates to divide complex numbers: (a+bi)/(c+di) = (a+bi)(c–di)/(c²+d²).
用共轭做复数除法：(a+bi)/(c+di) = (a+bi)(c–di)/(c²+d²)。
Visualise transformations with graph paper such as grids and circles.
用坐标纸画出网格和圆来可视化变换。

11. Common Misconceptions | 常见误解

Many students think that i is just a symbol and forget that i² = –1 drives algebraic simplification. This leads to errors in multiplication.

很多学生以为 i 只是一个符号，忘记了 i² = –1 是代数化简的关键，从而导致乘法出错。

Another mistake is treating complex functions like real ones when it comes to graphs – one cannot simply draw y=f(x) curves.

另一个错误是在画图时把复变函数当作实函数，想当然地绘制 y=f(x) 曲线，这是不行的。

A function can be continuous but not differentiable in the complex sense (e.g., f(z)=z̄). Differentiability in complex analysis is much stricter.

一个函数可能连续，但在复数意义上不可导（如 f(z)=z̄）。复分析中的可微性条件要严格得多。

12. Summary of Key Points | 考点总结

A complex function f maps complex numbers to complex numbers. It can be broken into real functions u and v. Simple functions like z, z+c, z², and 1/z reveal rich transformations: translation, rotation, scaling, inversion, and angle doubling. Modulus and argument are powerful tools to understand these effects. The concept of continuity paves the way for future study of holomorphic functions, which are the heart of complex analysis.

复变函数 f 将复数映射为复数，它可以分解为实函数 u 和 v。简单的函数如 z、z+c、z² 和 1/z 揭示了丰富的变换：平移、旋转、伸缩、反演和角度加倍。模和辐角是理解这些效果的有力工具。连续性的概念为进一步研究全纯函数铺平了道路，而全纯函数正是复分析的核心。

Even at KS3 level, playing with these ideas builds algebraic confidence and nurtures mathematical curiosity. Keep practicing the separation into real and imaginary parts, and soon you will be ready for more advanced wonders like the Julia sets.

即使在KS3阶段，摆弄这些概念也能建立代数信心并培养数学好奇心。坚持练习将复变函数分离为实部和虚部，很快你就能迎接更高级的奇观，比如朱利亚集。

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Complex Functions: Key Points | 复变函数：考点精讲