Mathematics of Physics and Engineering

Power Series and ODEs 231

symmetric interval is zero. FOR EXAMPLE,

./_«, 1 + 2x^6 + 5x^8

even though some computer algebra systems do not recognize this. Second,
if the original integral is not over the region we want, but otherwise is of the
suitable type, we can try and use symmetry to extend the integration to
the large interval. One basic example is / 0 °° F(x)dx = (1/2) J^° F(x)dx,
if the function F is even, that is, F(x) = F(—x). Still, as (4.4.21) shows,
sometimes it is better to keep the original interval. More subtle symmetry
can happen with trigonometric functions. FOR EXAMPLE, verify that

F* 1 — cos <p 1 r2v 1 — cos <p •K
J 0 5 + 3cos<^ V ~ 2 JQ 5 + 3cosy> ^ ~~ 3'
There exist many other classes of real integrals that are evaluated using
residues, but most of the examples include roots and natural logarithms.
Evaluation of such integrals relies on the theory of multi-valued analytic
functions and is discussed in a special Complex Analysis course.

4.4.3 Power Series and Ordinary Differential Equations

Many problems in mathematics, physics, engineering, and other sciences
are reduced to a linear second-order ordinary differential equation (ODE)

A(x)y"(x) + B(x)y'(x) + C(x)y(x) = 0. (4.4.22)

Even though the solution of such equations is usually not expressed in terms
of elementary functions, a rather comprehensive theory exists describing
various properties of the solution. This theory is based on power series and
was developed in the second half of the 19th century by the German math-
ematicians LAZARUS FUCHS (1833-1902) and GEORG FROBENIUS (1849-
1917). In what follows, we will outlines the main ideas of the theory; we
will need the results later in the study of partial differential equations.

We assume that the functions A, B, C can be extended to the complex
plane, and, instead of (4.4.22), we consider the equation in the complex
domain

w"{z) + p(z)w'(z) + q(z)w(z) = 0 (4.4.23)

for the unknown function w of the complex variable z, where p(z) =

B{z)/A(z), q(z) = C(z)/A(z).