Page 79 - Calculus with Complex Numbers

Ffgure 7.2

There is also a singularity of tan z/zl at c = 0 where the Laurent expansion is
(see Section 3.6)

Therefore the residue at z = 0 is 1.
lt follows that

which gives

oo 1 :a.2
z = 4 '
X (2rl + 1)

equivalently,
1 1 1 ,r2
1 + --i + -y + --i + ' ' ' = - ,
3 5 7 8
provided we can show

/
.)

as N -> co.
For this it is suflicient as previously to show tan z is bounded on yx for all N .
W hich it is since

sinc 2 sin2 x + sinhz y 1 + sinhz n'
I tan zI2 = = . s = cothz n'
cos c cos2 x + sinhz y sinhz n.
.