Complex Analysis

1. Applications of Complex Analysis
   1. Fluid Dynamics
      1. Complex Potential Functions
         1. Definition and Examples
         2. Applications to Irrotational Flows
         3. Use in Flow around Obstacles
      2. Conformal Mapping Techniques
         1. Transformation of Flow Problems
         2. Mapping Between Different Geometries
         3. Example: Flow around an Airfoil
      3. Vortex Methods
         1. Generation of Singularities
         2. Analysis of Vortex Motion
         3. Interaction of Vortices in a Fluid
      4. Stream Functions
         1. Analytic Representation
         2. Relation to Velocity Fields
         3. Incompressible Flow Analysis
   2. Quantum Physics
      1. Schrödinger Equation Solutions
         1. Complex Wave Functions
         2. Analytical Techniques for Quantum Wells
         3. Quantum Harmonic Oscillator
      2. Path Integrals
         1. Complex Action Functionals
         2. Applications in Quantum Field Theory
         3. Evaluation Techniques using Complex Methods
      3. Complex Probability Amplitudes
         1. Interpretation in Quantum Mechanics
         2. Relation to Probability Densities
         3. Physical Implications of Complex Phase
   3. Signal Processing and Control Theory
      1. Laplace Transforms and Complex Frequency
         1. Convergence Issues and Poles
         2. Applications in System Analysis
         3. Stability Analysis in Control Systems
      2. Fourier Transforms
         1. Signal Representation in the Complex Plane
         2. Applications to Signal Filtering
         3. Analysis of Band Pass Filters
      3. Bode Plots and Nyquist Criteria
         1. Stability Margins and Complex Analysis
         2. Frequency Response Representation
         3. Complex Gain and Phase Angle Analysis
      4. Pole-Zero Analysis
         1. Design of Linear Filters
         2. Feedback System Analysis
         3. Understanding Resonant Frequencies
   4. Number Theory and Zeta Functions
      1. Riemann Zeta Function
         1. Analytic Properties and Singularities
         2. Critical Line Hypothesis
         3. Applications in Prime Number Theorems
      2. Dirichlet L-functions
         1. Analytic Continuation
         2. Applications in Distribution of Prime Numbers
         3. Use in Modular Forms
      3. Modular Functions and Forms
         1. Elliptic Functions in Number Theory
         2. Applications to Partition Functions
         3. Complex Multiplication and its Arithmetic Significance
      4. Applications of Entire Functions
         1. Value Distribution Theory
         2. Use in Solving Diophantine Equations
         3. Application in Transcendental Number Theory