Overview
This lecture explains the mathematics behind the complex Fourier series, showing how functions can be broken down into sums of rotating vectors (complex exponentials) to model phenomena like heat flow and draw complex shapes.
Fourier Series and Rotating Vectors
- A complex Fourier series represents functions as sums of rotating vectors, each with an integer frequency.
- By adjusting the amplitude and angle of each vector, you can approximate any shape with enough vectors.
- Each vector's rotation is described by ( e^{i n 2\pi t} ), where ( n ) is an integer frequency.
Application to Heat Equation
- The heat equation describes temperature evolution along a rod and is linear (the sum of solutions is also a solution).
- If the initial temperature is a cosine wave, the solution remains a cosine wave with exponential decay.
- Combining exponentially decaying waves with different frequencies models complex initial temperature distributions.
- Over time, higher frequency terms decay faster, smoothing the temperature distribution.
Infinite Sums and Function Approximation
- Any function, even discontinuous ones like step functions, can be approximated as an infinite sum of sine or cosine waves (its Fourier series).
- Finite sums only yield approximations; the true function appears in the infinite sum (limit).
- At discontinuities, the series converges to the average of the left and right limits.
Finding Fourier Coefficients
- Each Fourier coefficient ( c_n ) determines the initial size and direction of a vector and is found by integrating the function multiplied by ( e^{-i n 2\pi t} ).
- The integral effectively "selects" the desired frequency component by cancelling out all others.
Complex Functions and Exponentials
- Extending the idea to complex-valued functions enables more general and visually intuitive decompositions (e.g., drawing shapes in the plane).
- Complex exponentials naturally correspond to rotation on the unit circle and simplify computations.
Numerical Computation and Visualization
- In practice, Fourier coefficients are computed numerically by summing over small intervals.
- SVG file data can be used to map a parameter ( t ) to points in space for visualization.
Key Terms & Definitions
- Fourier Series — Representation of functions as infinite sums of sines/cosines or complex exponentials.
- Complex Exponential ( e^{i\theta} ) — Describes rotation around the unit circle in the complex plane.
- Frequency — Number of complete rotations per unit interval, an integer ( n ) in this context.
- Coefficient ( c_n ) — Complex constant determining the amplitude and phase of each frequency component.
- Linear Equation — An equation where the sum of solutions is also a solution.
- Step Function — A function that jumps from one value to another, such as modeling two rods at different temperatures.
Action Items / Next Steps
- Compute Fourier coefficients for a step function by integrating as described.
- Relate the complex exponential form of Fourier series to the traditional sine/cosine form.
- (Optional) Read/watch additional materials on Fourier series for deeper understanding.