Unlock the concepts behind complex numbers

Applications of Complex Numbers

Physics

• Quantum Mechanics: \( \psi(x) = Ae^{ikx} \)
• Electrodynamics: \( Z = R + iX \)
• Wave Mechanics: \( \Psi(x,t) = Ae^{i(kx – \omega t)} \)

Engineering

• Electrical Engineering: \( V(t) = Ae^{i\omega t} \)
• Control Theory: \( F(s) = \frac{N(s)}{D(s)} \), where \( s = \sigma + i\omega \)
• Mechanical Vibrations: \( x(t) = A\cos(\omega t) + iB\sin(\omega t) \)
• Fluid Dynamics: Navier-Stokes: \( i\omega u – \nu \nabla^2 u = -\nabla p \)
• Telecommunications: \( S(t) = Ae^{i2\pi f t} \)

Computer Science

• Image Processing: \( F(u,v) = \int\int f(x,y)e^{-i2\pi(ux+vy)}dxdy \)
• Machine Learning: Complex-valued neural networks
• Data Compression: \( X(k) = \sum_{n=0}^{N-1} x(n)e^{-i2\pi kn/N} \)

Mathematics

• Calculus: \( \oint_C f(z)dz \)
• Differential Equations: \( ay” + by’ + cy = Ce^{i\omega t} \)
• Number Theory: Gaussian Integers \( a+bi \)
• Linear Algebra: \( Ax = \lambda x \), \( \lambda = a + bi \)
• Optimization: \( z = re^{i\theta} \)

Economics

• Game Theory: Payoff matrices involving \( i \)
• Financial Modeling: \( S(t) = S(0)e^{(r – i\sigma^2)t} \)

Geography

• Seismology: Wave equations with \( i \)
• Oceanography: \( \eta(x,t) = Ae^{i(kx – \omega t)} \)

Medicine

• MRI Technology: \( M(x) = \int\rho(x)e^{-ikx}dx \)
• Biomedical Engineering: \( H(s) = \frac{Y(s)}{X(s)} \) where \( s = \sigma + i\omega \)

Other Applications

• Music: \( f(t) = A_1\sin(2\pi f_1t) + iA_2\sin(2\pi f_2t) \)
• Astronomy: Celestial mechanics involving \( i \)
• Cryptography: \( E(x) = (ax + bi) \mod m \)
• Robotics: \( Z = x + iy \) for control systems