Complex Variables and Application

Author: Xuan Dai

• Chapter1 Complex Numbers

  • Sums and Products

    • Complex numbers can be defined as ordered pairs (x, y) of real numbers that are to be interpreted as points in the complex plane.
      

    • $z = (x, y) \ \ \ \ x = Re z, y = Im z$
    • $z_1=z_2\ iff\ Re\ z_1 = Re\ z_2\ and\ Im\ z_1 = Im\ z_2$
    • Sum: $(x_1,\ y_1) + (x_2,\ y_2) = (x_1 + x_2,\ y_1 + y_2)$
    • Product: $(x_1,\ y_1)(x_2,\ y_2) = (x_1x_2 - y_1y_2,\ y_1x_2 + x_1y_2)$

  • Basic Algebraic Properties

    • Commutative Laws: $z_1 + z_2 = z_2 + z_1,\ z_1z_2=z_2z_1$
    • Associative Laws: $(z_1 + z_2) + z_3=z_1 + (z_2 + z_3),\ (z_1z_2)z_3=z_1(z_2z_3)$
    • Distribute Laws: $z(z_1 + z_2) = zz_1 + zz_2$
    • The additive identity is 0 = (0, 0), and the multiplicative identity is 1 = (1, 0).
    • z + 0 = z and z * 1 = z
    • Addtive inverse: -z = (-x, -y) which satisfies z + (-z) = 0
    • Multiplicative inverse: $z^{-1} = (\frac{x}{x^2+y^2},\ \frac{-y}{x^2+y^2})$ which satisfies $z*z^{-1}$ = 1

  • Further Properties

    • If $z_1z_2 = 0$, then at least one of the factors z1 and z2 is 0.
    • Subtraction: $z_1 - z_2 = z_1 + (-z_2)$
    • Division: $$\frac{z_1}{z_2} = z_1z^{-1}\ \ (z_2 \ne 0)$$
    • $z_1 - z_2 = (x_1-x_2,\ y_1-y_2)$
    • $$\frac{z_1}{z_2} = (\frac{x_1x_2+y_1y_2}{x_2^2+y_2^2},\ \frac{y_1x_2-x_1y_2}{x_2^2+y_2^2})$$
    • An easy way to remember: $$\frac{z_1}{z_2} = \frac{(x_1+iy_1)(x_2-iy_2)}{(x_2+iy_2)(x_2-iy_2)}$$
      Note Note that $(x_2+iy_2)(x_2-iy_2) = x_2^2 + y_2^2 \in R$
    • $$\frac{z_1+z_2}{z_3} = (z_1+z_2)z_3^{-1} = z_1z_3^{-1} + z_2z_3^{-1} = \frac{z_1}{z_3} + \frac{z_2}{z_3}$$
    • $$\frac{1}{z}=z^{-1}$$
    • $$\frac{z_1}{z_2} = z_1(\frac{1}{z_2})$$
    • $$(\frac{1}{z_1})(\frac{1}{z_2}) = z_1^{-1}z_2^{-1} = (z_1z_2)^{-1} = \frac{1}{z_1z_2}$$
    • $$(\frac{z_1}{z_3})(\frac{z_2}{z_4})=\frac{z_1z_2}{z_3z_4}\ \ (z_3\neq 0,\ z_4\neq 0)$$
    • $$(z_1+z_2)^n = \sum_{k=0}^{n}C_n^kz_1^kz_2^{n-k}$$

  • Vectors and Moduli

    • Any complex number is associated a vector from the origin to the point (x, y)
      

    • $|z| = \sqrt{x^2 + y^2}$ and $|z_1-z_2| = \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}$
      

    • $|z|^2 = (Re\ z)^2 + (Im\ z)^2$
    • $Re\ z \le |Re\ z| \le |z|$ and $Im\ z \le |Im\ z| \le |z|$
    • Triangle inequality: $|z_1+z_2| \le |z_1|+|z_2|$
    • $|z_1+z_2| \ge ||z_1|-|z_2||$
    • $|z_1+z_2| \ge |z_1|-|z_2|$
    • $||z_1|-|z_2|| \le |z_1 \pm z_2| \le |z_1|+|z_2|$

  • Complex Conjugates(共轭负数)

    • $\bar{z} = x-iy$
      

    • $\bar{z_1+z_2} = \bar{z_1} + \bar{z_2}$
    • $\bar{z_1-z_2} = \bar{z_1} - \bar{z_2}$
    • $\bar{z_1z_2} = \bar{z_1}\bar{z_2}$
    • $\bar{(\frac{z_1}{z_2})} = \frac{\bar{z_1}}{\bar{z_2}} \ \ \ (z_2 \neq 0)$
    • $Re\ z = \frac{z+\bar{z}}{2}\ \ and\ \ Im\ z = \frac{z-\bar{z}}{2i}$
    • $z\bar{z} = |z|^2$
    • $|z_1z_2| = |z_1||z_2|\ \ \Rightarrow\ |z^n|=|z|^n$
    • $|frac{z_1}{z_2}| = frac{|z_1|}{|z_2|}\ \ \ (z_2 \neq 0)$

  • Exponential Form

    • Polar form: z = r(cosθ+isinθ)
    • arg z = Arg z + 2nπ (arg z: 辐角$\theta$, Arg z: 辐角主值$\Theta$)
      

    • Euler's formula: $e^{i\theta} = cos\theta + isin\theta$ (可用泰勒展开式证明~)
    • $z = re^{i\theta}$
      

    • $z = Re^{i\theta}$ and $z = z_0 + Re^{i\theta}$
      

  • Products and Powers in Exponential Form

    • $z_1z_2 = r_1e^{i\theta_1}r_2e^{i\theta_2} = r_1r_2e^{i\theta_1}e^{i\theta_2} = (r_1r_2)e^{i(\theta_1+\theta_2)}$
    • $\frac{z_1}{z_2} = \frac{r_1}{r_2}e^{i(\theta_1-\theta_2)}$
    • $z^{-1} = \frac{1}{r}e^{-i\theta}$
    • $z^n = r^ne^{in\theta}$
    • $(e^{i\theta})^n = e^{i\theta n}$
    • The de Moivre's formula: $(cos\theta+isin\theta)^n = cosn\theta + isinn\theta$

  • Arguments of Products and Quotients

    • $z_1z_2 = (r_1r_2)e^{i(\theta_1+\theta_2)}$
    • $arg(z_1z_2) = arg\ z_1+arg\ z_2$
      

    • $arg\ (z_2^{-1}) = -arg\ z_2$
    • $arg(\frac{z_1}{z_2}) = arg\ z_1 - arg\ z_2

  • Roots of Complex Numbers
    

    • If two nonzero complex numbers are equal iff r1 = r2 and $θ_1 = θ_2 + 2kπ$
      

    • The nth roots of $z_0$ are $$z = \sqrt[n]{r_0}exp[i(\frac{\theta_0}{n}+\frac{2k\pi}{n})],\ \ k = 0,\ \pm1,...$$
      
      
      

  • Examples

  • Regions in The Complex Plane

    • $\varepsilon$ neighborhood
      

    • interior point: A point z is said to be an interior point of a set S whenever there is some neighborhood of z that contains only points of S. (只要有一个就够了！)
    • exterior point: A point z is said to be an exterior point of a set S when there exists a neighborhood of it containing no points of S. (同样一个就够了！)
    • boundary point: Neither interior nor exterior. Therefore, a boundary point is a point all of whose neighborhoods contain at least one point in S and at least one point not in S.
      

    • open set: A set is open if it and only if each of its points is an interior point.
    • closed set: A set is closed if it contains all of its boundary points.
    • closure of a set: The closure of a set S is the closed set consisting of all points in S together with the boundary of S.
      

    • connected: An open set S is connected if each pair of points z1 and z2 in it can be joined by a polygonal line, consisting of a finite number of line segments joined end to end, that lies entirely in S.
      

    • domain: An open connected set. (any neighborhood is a domain)
    • region: A domain together with some, none, or all of it boundary points is referred to as a region.
    • bounded: A set S is bounded if every point of S lies inside some circle |z|=R. Otherwise, it is unbounded.

    • accumulation point: A point 0 is said to be an accumulation point of a set S if each deleted neighborhood of z contains at least one point of S.

      • If a set S is closed, then it contains each of its accumulation points.
      • A set is closed iff it contains all of its accumulation points.
      • An Interior point must be an accumulation point.
      • An Exterior point must not be an accumulation point.