244 Chapter 8Complex numbers
We note that the quantization of the energy of the system has arisen as a consequence
of applying the periodicity condition (periodic boundary condition) to the solutions.
We note also that the set of wave functions includes all the functions (8.45) for the
possible periodicities around a circle.
0 Exercise 48
8.7 Evaluation of integrals
Integration with respect to a complex variable is an important part of the theory
of functions of a complex variable, but is used only in advanced applications in the
physical sciences. Ordinary integration over complex functions obeys the same
rules as integration over real functions. In addition, complex functions can be used
to simplify the evaluation of certain types of integral. For example, it is shown in
Example 6.13, how the integral
can be evaluated by the method of integration by parts. An alternative, more elegant,
method is to express the trigonometric function in terms of the (complex) exponential
function. We consider the general form
Because cos 1 bxis the real part ofe
ibxit follows that the integralIis the real part of
the integral obtained fromIby replacing cos 1 bxbye
ibx:
The complex integral is evaluated by means of the ordinary rule for the integration of
an exponential function. Thus (ignoring the constant of integration),
and this can be resolved into its real and imaginary parts:
=
++ −
e
ab
a bx b bx i a bx b bx
ax()
(cos sin ) (sin cos )
22
Zedxe
bx i bx
aib
()aibx axcos sin
+=
Zedx
e
aib
e
e
aib
aibxaibxaxibx()()++=
=
Ieedxedx
ax ibx a ib x==
+ReZZRe
()I e bx dx
ax=Z cos
Zexdx
−axcos