We
continue now looking at more examples of complex integration and application of
the residue theorem.

Example 1:

∫

_{-}_{¥}^{¥}^{ }(1 + x^{2}) dx**/**1 + x^{4}
Find
the residues at exp(pi/4) and exp(3pi/4)
and hence integrate the preceding:

Solution:

__singular po__ints where:

f(z)
= 1 + x

^{2}**/**1 + x^{4}
Now: We note that
1 + x

^{4}= 0 when x = (-1)^{¼}^{ }
Or
the fourth root of (-1). The modulus of (-1) = 1 and the argument is p. Therefore,
by

*de Moivre’s theorem*:
(-1 + 0i)

^{ }^{¼ }^{ }=^{4}Ö(-1) [ cos (p + 2kp/ 4) + i sin (p + 2kp/ 4)]
k= 0, 1, 2, 3…….

At
k= 0: (-1)

^{¼}^{ }= cis (p/4) = exp (pi/4)
At
k=1: (-1)

^{¼}^{ }= cis (3p/4) = exp (3pi/4)(We’re not concerned with k=2, 3……Why?)

We
therefore take the semi-circular contour as shown in the graphic. So that:

**ò**

_{C}_{R}

^{ }f(z) dz + ∫

_{-R}

^{R}

^{ }f(x) dx = 2 pi (sum of residues inside C)

As R increases without bound we have:

∫

2 pi [Res f(z) at exp (pi/4) and exp (3pi/4) ]

_{-}_{¥}^{¥}^{ }(1 + x^{2}) dx**/**1 + x^{4}=2 pi [Res f(z) at exp (pi/4) and exp (3pi/4) ]

For
the case f(z) = p(z)/ q(z) Then:

Res
f(z)

**= lim**_{ z = a}_{ z }**[p(z)/ q’(z)]**_{® a}
Then
for z = exp (pi/4):

Res
f(z)

_{z = exp }**= lim**_{pi/4}_{ z }_{® exp }**1 + z**_{pi/4}^{2}**/**4 z^{3}=
1
+ exp(pi/2) / 4 exp(3pi/4) = 1 + cis (p/2) / 4 cis (3p/4)

Res
f(z)

_{z = exp }**= 1 + i / 4(- Ö2/ 2 + iÖ2/ 2 )**_{pi/4}
= 1/ 2

**Ö**2 (i+ 1/ i- 1)
Rationalize
the denominator to get: 2i/ -2 = -

**Ö**2i/ 4
Then
for z = exp (3pi/4):

Res
f(z)

_{z = exp 3}**= lim**_{pi/4}_{ z }_{® exp 3}**1 + z**_{pi/4}^{2}**/**4 z^{3}=
1
+ exp(3pi/4) / 4 exp(9pi/4) =

¼
[exp(-3pi/4) +
exp(- 9pi/4)]
=

¼ [ cis (-3pi/4) + cis (- 9pi/4)]

¼ [ cis (-3pi/4) + cis (- 9pi/4)]

Therefore:

∫

2 pi [Res f(z) at exp (pi/4) and exp (3pi/4) ]

_{-}_{¥}^{¥}^{ }(1 + x^{2}) dx**/**1 + x^{4}=2 pi [Res f(z) at exp (pi/4) and exp (3pi/4) ]

= 2 pi {-

**Ö**2i/ 4 + ¼ [ cis (-3pi/4) + cis (- 9pi/4)]}__Problem for Serious Math Mavens__:

Obtain
the integral for:

∫

_{-}_{¥}^{¥}^{ }x dx**/**(x**- 2x + 2)**^{2}
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