Complex Analysis

Complex Numbers

• A complex number is a number of the form z = x + yi, where x and y are real numbers and i is the imaginary unit, satisfying i^2 = -1.
• The set of complex numbers is denoted by ℂ.
• The real and imaginary parts of a complex number z are denoted by Re(z) and Im(z), respectively.

Functions of Complex Numbers

• A function f(z) of a complex variable z is a function that maps each complex number z to a complex number w.
• The function f(z) is said to be analytic at a point z_0 if it is differentiable at z_0 and in a neighborhood of z_0.
• Cauchy-Riemann Equations: A function f(z) = u(x, y) + iv(x, y) is analytic at z_0 if and only if it satisfies the Cauchy-Riemann equations:
  • ∂u/∂x = ∂v/∂y
  • ∂u/∂y = -∂v/∂x

Contour Integration

• A contour is a piecewise-smooth curve in the complex plane.
• The contour integral of a function f(z) over a contour C is denoted by ∫_C f(z) dz.
• Cauchy's Integral Formula: If f(z) is analytic inside and on a contour C, and z_0 is a point inside C, then:
  • f(z_0) = (1/2πi) ∫_C f(z)/(z-z_0) dz

Residue Theory

• The residue of a function f(z) at a point z_0 is denoted by Res(f, z_0).
• The residue theorem: If f(z) is analytic inside and on a contour C, except at a point z_0 inside C, then:
  • ∫_C f(z) dz = 2πi Res(f, z_0)
• The residue of a function f(z) at a pole of order n can be calculated using:
  • Res(f, z_0) = (1/(n-1)!) lim_(z→z_0) (d^(n-1)/dz^(n-1))((z-z_0)^n f(z))

Applications of Complex Analysis

• Complex analysis has applications in:
  • Electrical engineering (e.g., circuit analysis)
  • Signal processing (e.g., Fourier transform)
  • Quantum mechanics (e.g., wave functions)
  • Number theory (e.g., prime number theorem)

复分析

复数

• 复数是形如 z = x + yi 的数，其中 x 和 y 是实数，i 是虚单位，满足 i^2 = -1。
• 复数的集合记为 ℂ。
• 复数 z 的实部和虚部分别记为 Re(z) 和 Im(z)。

复变函数

• 复变函数 f(z) 是将每个复数 z 映射到另一个复数 w 的函数。
• 函数 f(z) 在点 z_0 是可微分的当且仅当它在 z_0 和 z_0 的邻域内是可微分的。
• 柯西-黎曼方程：函数 f(z) = u(x, y) + iv(x, y) 在 z_0 是可微分的当且仅当它满足柯西-黎曼方程：
  • ∂u/∂x = ∂v/∂y
  • ∂u/∂y = -∂v/∂x

积分 along a contour

• 剖面是复平面中的 piecewise-smooth 曲线。
• 函数 f(z) 沿着剖面 C 的积分记为 ∫_C f(z) dz。
• 柯西积分公式：如果 f(z) 在剖面 C 内部和边界上是可微分的，并且 z_0 是 C 内部的一点，那么：
  • f(z_0) = (1/2πi) ∫_C f(z)/(z-z_0) dz

残余理论

• 函数 f(z) 在点 z_0 的残余记为 Res(f, z_0)。
• 残余定理：如果 f(z) 在剖面 C 内部和边界上是可微分的，除了点 z_0 内部，那么：
  • ∫_C f(z) dz = 2πi Res(f, z_0)
• 如果函数 f(z) 在点 z_0 有阶数为 n 的极点，那么可以使用以下公式计算残余：
  • Res(f, z_0) = (1/(n-1)!) lim_(z→z_0) (d^(n-1)/dz^(n-1))((z-z_0)^n f(z))

复分析应用

• 复分析在以下领域有应用：
  • 电气工程（例如，电路分析）
  • 信号处理（例如，傅里叶变换）
  • 量子力学（例如，波函数）
  • 数论（例如，素数定理）