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A double Fourier series is a way to represent a function of two variables as an infinite sum of sine functions. This concept extends the idea of a single-variable Fourier series to functions depending on two parameters, often denoted as \( x \) and \( y \). In this context, each term in the series takes the form of a product of sine functions: \( \sin \left( \frac{m\pi x}{a} \right) \sin \left( \frac{n\pi y}{b} \right) \). Here, \( m \) and \( n \) are positive integers representing the harmonic frequencies along the \( x \) and \( y \) directions respectively. The main idea is to express the target function \( f(x, y) \) as a superposition of these basis functions. The double Fourier series for \( f(x, y) \) is given by the expression:

• \( \sum_{m=1}^{\infty} \sum_{n=1}^{\infty} C_{mn} \sin\left(\frac{m\pi x}{a}\right) \sin\left(\frac{n\pi y}{b}\right) \)

Each coefficient \( C_{mn} \) quantifies how much of the corresponding sine product is needed to construct the function. This series is incredibly useful for solving partial differential equations and analyzing periodic functions on bounded rectangular domains.

The calculation of coefficients \( C_{mn} \) in a double Fourier series relies heavily on integration techniques. For each coefficient, the process involves evaluating a double integral. The given formula:

• \( C_{mn} = \frac{4}{ab} \int_0^a \int_0^b f(x, y) \sin\left(\frac{m\pi x}{a}\right) \sin\left(\frac{n\pi y}{b}\right) \, dy \, dx \)

requires breaking it down into two separate integrals.
First, we tackle the integral over \( y \):

• \( \int_0^b y(b^2-y^2) \sin\left(\frac{n\pi y}{b}\right) \, dy \)

This typically requires techniques such as integration by parts or using trigonometric identities.Once the inner integral is solved, its result is used in the outer integral over \( x \):

• \( \int_0^a x(a^2-x^2) \cdot \text{Result from Step 3} \sin\left(\frac{m\pi x}{a}\right) \, dx \)

Again, this may also use integration by parts. These techniques allow us to simplify and evaluate seemingly complex integrals within the domain limits.

The coefficients \( C_{mn} \) play a critical role in constructing the double Fourier series. These coefficients are calculated by integrating the product of the function \( f(x, y) \) with the sine basis functions. Each \( C_{mn} \) is found using:

• \( C_{mn} = \frac{4}{ab} \int_0^a \int_0^b f(x, y) \sin\left(\frac{m\pi x}{a}\right) \sin\left(\frac{n\pi y}{b}\right) \, dy \, dx \)

The process involves:

• Substituting \( f(x, y) \) into the integral.
• Performing the integration over \( y \) first to simplify the expression.
• Using the result to evaluate the integral over \( x \).

After calculating these integrals, each coefficient tells us how much of each sine product is needed to represent the function at hand as accurately as possible. By combining all coefficients with their respective sine terms, we can perfectly reconstruct or approximate the original function within a desired boundary. This is particularly valuable in applications like signal processing, heat distribution, and vibration analysis.