Page 74 - Calculus with Complex Numbers

Su m m atio n of series

'7'. I Re si d ue s of cot az
Hlemental'y theol'y of sequences and series only allows vel'y few series to be
summed exactly. ln most cases one has to be content with knowing that a series
converges without knowing what the sum is. lt is however possible to sum a wide
class of series by exploiting properties of the complex cotangent function cot z.
The singularities of cot z = cos z/ sin z occur at the zeros of sin z which are at
z = n:r for integral n (see Section 2.9). The residues at these singularities can be
obtained by differentiating the denominator rule and are

R cos z cos z
es cot z = Res = = 1.
zuuznx zuuznx sin c cos c C=/X

'7'.2 Lau re nt expansi o n of cot az
We can either divide the Maclaurin exparlsions of cos z, sin z (as we did in
Section 3.6 for tan z) or use the expansion of tan z to obtain

7.3 The m ethod
W e demonstrate the method by summing the series

p
Z
)
oo 1
1