Transform Methods in Mathematics

  1. Advanced Applications
    1. Differential Equations
      1. Ordinary Differential Equations
        1. Linear ODEs with Constant Coefficients
          1. Initial Value Problems
            1. Boundary Value Problems
              1. Systems of ODEs
              2. Partial Differential Equations
                1. Heat Equation Solutions
                  1. Wave Equation Solutions
                    1. Laplace Equation
                      1. Transform Methods for PDEs
                      2. Green's Functions
                        1. Construction via Transforms
                          1. Applications to Boundary Value Problems
                        2. Systems and Signal Processing
                          1. Linear Time-Invariant Systems
                            1. Impulse Response
                              1. Transfer Functions
                                1. Frequency Response
                                2. Filter Design
                                  1. Analog Filters
                                    1. Digital Filters
                                      1. Filter Specifications
                                      2. Sampling and Reconstruction
                                        1. Sampling Theorem
                                          1. Aliasing
                                            1. Interpolation and Reconstruction
                                            2. Modulation Theory
                                              1. Amplitude Modulation
                                                1. Frequency Modulation
                                                  1. Digital Modulation
                                                2. Control Theory Applications
                                                  1. System Analysis
                                                    1. Stability Criteria
                                                      1. Root Locus Method
                                                        1. Frequency Domain Analysis
                                                        2. Controller Design
                                                          1. PID Controllers
                                                            1. Lead-Lag Compensation
                                                              1. State Feedback
                                                              2. Frequency Response Methods
                                                                1. Bode Plots
                                                                  1. Nyquist Plots
                                                                    1. Gain and Phase Margins
                                                                  2. Probability and Statistics
                                                                    1. Characteristic Functions
                                                                      1. Definition and Properties
                                                                        1. Moment Generation
                                                                          1. Distribution Identification
                                                                          2. Moment Generating Functions
                                                                            1. Relationship to Laplace Transform
                                                                              1. Applications in Statistics
                                                                              2. Random Process Analysis
                                                                                1. Power Spectral Density
                                                                                  1. Correlation Functions
                                                                                    1. Filtering of Random Signals
                                                                                  2. Image and Multidimensional Processing
                                                                                    1. Two-Dimensional Transforms
                                                                                      1. 2D Fourier Transform
                                                                                        1. 2D Discrete Transforms
                                                                                        2. Image Processing Applications
                                                                                          1. Image Enhancement
                                                                                            1. Image Compression
                                                                                              1. Feature Detection
                                                                                              2. Multidimensional Extensions
                                                                                                1. Higher Dimensional Transforms
                                                                                                  1. Separable Transforms

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                                                                                              5. Specialized Integral Transforms

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                                                                                              7. Computational Aspects and Numerical Methods