Consider expansion of a function in terms of a series \(F_{n}(x)\) in the following form:

\[f(x)=\sum_{n} A_{n} F_{n}(x)\]

If the functions \(F_{n}\) 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 \(F_{n}\) and \(F_{m}\).

Using this, show that the series coefficients can be calculated as

\[A_{n}=\frac{\int_{0}^{1} f(x) F_{n}(x) d x}{\int_{0}^{1} F_{n}^{2}(x) d x}\]

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 \(\cos [(n+1 / 2) \pi x]\) is orthogonal in the interval from zero to one. Hence verify the following expression for the series coefficient given in the text for \(A_{n}\) :

\[A_{n}=\frac{2(-1)^{n}}{(n+1 / 2) \pi}\]

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.