Overview and Fundamental Ideas - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 8

A n = { 2 f e (x) cos n x/L dx}/L (1.10)
Alternatively, if f(x) is assumed to be odd with f(x) = - f(-x), we can write

f o (x) = B n sin n x/L (1.11)

B n = { 2 f o (x) sin n x/L dx}/L (1.12)
In the above, the subscripts e and o indicate even and odd . Also, in Equations
1.9 to 1.12, the limits of integration extend from x = 0 to L, and the summation
extends from n = 1 to (limits are omitted for brevity).
Consider any function g(x) defined on the wider interval -L < x < L. From
g(x) = {g(x) + g(-x)}/2 + {g(x) - g(-x)}/2, it is clear that the first term {g(x) +
g(-x)}/2 remains unchanged, and that the second term {g(x) - g(-x)}/2 reverses
in sign, if in each case x is replaced by -x. Since {g(x) + g(-x)}/2 is even , while
{g(x) - g(-x)}/2 is odd , it follows from the above arguments that

g(x) = C 0 + C n cos n x/L + D n sin n x/L (1.13)

C 0 = { {g(x) + g(-x)}/2 dx}/L (1.14)

C n = { 2 {g(x) + g(-x)}/2 cos n x/L dx}/L (1.15)

D n = { 2 {g(x) - g(-x)}/2 sin n x/L dx}/L (1.16)

Examples of commonly used Taylor and Fourier series may be found in
mathematical tables and references.
1.3 Separation of Variables and Eigenfunction Expansions
Here we demonstrate how Fourier series are used to solve boundary value
problems. As suggested, models such as Equation 1.1 possess numerous
solutions, which must be fixed by auxiliary constraints called boundary and
initial conditions. We will emphasize constructive techniques, but not dwell
upon existence and uniqueness issues in this topic. Separation of variables is the
best known of all partial differential equation solution methods. Unfortunately,
for most purposes, it is also the least useful, requiring too many terms for
convergence; nonetheless, in many courses, it is the standard by which
comprehension and ingenuity are based. We introduce the technique using
examples marked by increasing levels of difficulty. We take the opportunity to
introduce the mathematical and engineering terminology used in solving
vibration problems. Our examples are defined on the finite domain 0 < x < L.