dwsmith said:$$
u_y(x,\pi) = \frac{x}{\pi} + \sum_{n = 1}^{\infty}nB_n\sin xn\cosh\pi n = 0.
$$
How can I solve for $A_n$ here?
A Fourier coefficient is a numerical value used in Fourier series to represent the contribution of a specific frequency component to a periodic function. It is calculated as the inner product of the function and a complex exponential function.
A_n represents the coefficient for the cosine term in the Fourier series. It is calculated using an integral formula involving the function and the cosine function with the corresponding frequency.
To solve for A_n, you must first determine the function and the corresponding frequency. Then, use the integral formula for A_n and evaluate the integral to find the numerical value of the coefficient.
This equation means that the function u(x,y) is equal to 0 at the point (x,π). This information can be used to solve for the Fourier coefficient A_n, as it provides a condition for the function at a specific point.
Fourier coefficients are important in scientific research because they allow us to represent complex periodic functions in terms of simpler trigonometric functions. This makes it easier to analyze and understand these functions, and can be applied to a wide range of fields such as signal processing, image and sound analysis, and quantum mechanics.