Page 96 - Calculus with Complex Numbers
P. 96
The integral vanishes for a outside y by Cauchy's theorem .
5 Evaluation of fi nite real i ntegrals
W e have
2
z
c
+
4
o
)
'
j
.
t
.
j
:
'
v
1
+
1:: dt 2 dz 2:)z.
j 2 s = - + c = /
by the residue theorem. The integrand has one singularity irlside y at z =
3
/- - 2
3
.$,. , where the residue is 1/2./- (differentiate the denominator).
W e have
2
+
1:: dt in t = Jy .:2 + 3i c - 1 = s,'-j '
s
j 3 dz 2:)z.
The integrand has one singularity inside y at z = (Vl$' - 3)ï/2, where the
residue is I/V'V .
W e have
2,, dt c dz
= 4/ = n'.
o 4 - 3 cos2 t r 3c4 - 10c2 + 3
The integrand has two singularities inside y at z = ulu 1/.$//X, where the
residues are lnoth equal to - 1/ 16.
w e have
z,c 1 1 6 dz
cos 6 t dt = - c + -
o 64 p' z iz
j
'
?
;
' j (c6 + 6c4 + lsc2 + 2o+ -p- + 6 + ' ) t'z - 5,,'
= & / 1 ( F ' 8
7
5
