Transform Methods in Mathematics

  1. Fourier Analysis
    1. Fourier Series
      1. Periodic Function Representation
        1. Trigonometric Fourier Series
          1. Sine and Cosine Expansions
            1. Coefficient Calculation
              1. Even and Odd Function Properties
              2. Complex Exponential Fourier Series
                1. Euler's Formula Applications
                  1. Complex Coefficients
                    1. Relationship to Trigonometric Form
                    2. Convergence Properties
                      1. Dirichlet Conditions
                        1. Pointwise Convergence
                          1. Uniform Convergence
                            1. Mean Square Convergence
                            2. Gibbs Phenomenon
                              1. Overshoot at Discontinuities
                                1. Practical Implications
                                2. Parseval's Identity for Series
                                  1. Energy Conservation
                                    1. Power Spectrum
                                  2. Fourier Transform
                                    1. Transition from Series to Transform
                                      1. Limiting Process
                                        1. Aperiodic Functions
                                        2. Definition and Existence
                                          1. Forward Transform
                                            1. Inverse Transform
                                              1. Existence Conditions
                                              2. Physical Interpretation
                                                1. Frequency Spectrum
                                                  1. Amplitude and Phase Spectra
                                                    1. Energy Spectral Density
                                                    2. Properties
                                                      1. Linearity
                                                        1. Time Shifting
                                                          1. Frequency Shifting
                                                            1. Time Scaling
                                                              1. Duality Property
                                                                1. Differentiation Properties
                                                                  1. Integration Properties
                                                                    1. Convolution Theorem
                                                                      1. Multiplication Theorem
                                                                        1. Parseval's Theorem
                                                                        2. Transform Pairs
                                                                          1. Rectangular Pulse
                                                                            1. Gaussian Function
                                                                              1. Exponential Functions
                                                                                1. Sinc Function
                                                                                  1. Delta Function
                                                                                2. Specialized Fourier Transforms
                                                                                  1. Fourier Sine Transform
                                                                                    1. Definition and Properties
                                                                                      1. Applications to Odd Functions
                                                                                      2. Fourier Cosine Transform
                                                                                        1. Definition and Properties
                                                                                          1. Applications to Even Functions
                                                                                          2. Relationship Between Transforms
                                                                                          3. Applications
                                                                                            1. Signal Analysis
                                                                                              1. Spectrum Analysis
                                                                                                1. Filtering
                                                                                                  1. Modulation
                                                                                                  2. Partial Differential Equations
                                                                                                    1. Heat Equation
                                                                                                      1. Wave Equation
                                                                                                        1. Boundary Value Problems

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                                                                                                    2. The Laplace Transform

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                                                                                                    4. Discrete Transform Methods