Page 78 - Calculus with Complex Numbers
P. 78
Hence we obtain
oo (- 1)n :7.,2
nl 12 '
1
equivalently,
1 1 1 :/2
1 - + - + . . . = 12
ï
x
f
l
k
è
as before.
7.6 U se of tan a
z
W e can sum the series
given that
by observing that
Alternatively, we can use the integral
tall z
2 dz ,
pk .'
!
where yx is the square centre at 0 with half side Nn'(Figure 7.2).
The singularities of tan z are at z = (rl + 1/2):v with residues - 1. Therefore
the residue of tan z/zl at c = (r! + 1/2):v is
tan z 1 4
Res =z - =z - .
,
z=(n+1/z),r .::2 (n + 1/2)2>2 (2r! + 1)2:7.2
