Consider expansion of a function in terms of a series Fn(x) in the following form:-f(x) = ∑n AnFn(x) If the functions Fn are orthogonal then this property helps to unfold the series and permits us to find the series coefficients, one at a time

Consider expansion of a function in terms of a series Fn(x) in the following form:-f(x) = ∑n AnFn(x) If the functions Fn are orthogonal then this property helps to unfold the series and permits us to find the series coefficients, one at a time. State what is meant by orthogonality of two functions Fn and Fm. Using this, show that the series coefficients can be calculated as

An = 0∫1 f(x)Fn(x)dx/ 0∫1 F2 n(x)dx where the domain of solution is assumed to be from zero to one. Apply this to the case of a slab with constant initial temperature (f(x) = 1). Show that the function cos[(n + 1/2)πx] is orthogonal in the interval from zero to one. Hence verify the following expression for the series coefficient given in the text for An:

An = 2(−1)ⁿ (n + 1/2)π for a constant initial condition equal to one. What is the series coefficient if the initial temperature is a linear function of position? Note that tools such as MATHEMATICA and MAPLE can be used to do the algebra and you may wish to use them.