Page 99 - Calculus with Complex Numbers
P. 99
Sum m ation of series
>4/90.
7>4/720, >4/96.
Observe that
CO j CO j j
X n (s + 1 ) - Y/ ( -s - n + 1 )
1
( l - .l. ) + ( ) - .l.) + . . . + ( ) - s ) 1 ) + . . . - l .
a
a
-
lf y is the square with centre 0, and half side N + 1/2, then
cot n'z 8 (N + 1/2) coth zr/2 16 coth zr/2
16 c(c + 1) dz Y (>F + 1/2)(>F + 3/2) - 2:7 + 3 -* 0
r
as N --> co. For n # 0 - 1 we have
cot n'z 1
Res = .
z=n c(c + 1) n'n (n + 1)
At z = 0 we have
which shows that z = 0 is a double pole with residue - 1/>.
At z = - 1 we have putting t = z + 1
which shows that z = - 1 is also a double pole with residue - 1/>.
Therefore by the residue theorem we have
N .V-1
-
+
n
n
)
1
F
-
1
+
)
L
y
z
(
n
n
1
+
'
)
/
J c ' - . - ( - . n -
cotz,c 1 x..-- 1 1 x..-- 1 1. '
c
(
'
1
&
.
n
Z
-
n
