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Complex Numbers and Functions

Source lecture(s): PHY622 Lec3

Intuition

Complex numbers extend the real line to a plane, turning many hard algebra problems into geometry and vice versa.

Formal Definition

A complex number \(z=x+iy\) has real part \(x\) and imaginary part \(y\). Polar form: \(z=re^{i\theta}\).

Mathematical Formulation

\[e^{i\theta}=\cos\theta+i\sin\theta$$ $$|z|=\sqrt{x^2+y^2}$$ $$\arg z = \tan^{-1}(y/x)\]

Derivation

Euler's formula follows from the Taylor series of \(e^x\), \(\sin x\), \(\cos x\).

Worked Example

\((1+i)^4 = ((1+i)^2)^2 = (2i)^2 = -4\).

Common Mistakes

  • Treating \(\arg z\) as single-valued without branch cuts.
  • Confusing \(z^*\) (conjugate) with \(1/z\).

Quiz

Q1. What is \(e^{i\pi}\)?

Answer

\(-1\).