Page 57 - Calculus with Complex Numbers

c - 2
.
(iii) dz :jq 4.$,/*9 where y is the square with vertices at ulu 1 ulu i .
r c - 3
Find all the singularities of the following functiorls. Use the method of
differentiating the denominator to lind all the residues.

z + 1 ez 1
(i) , (ii) , (iii) z .
z - 1 . :2 + :7r2 z - 6. + 8
:
Use the residue theorem to evaluate the following integrals round the contours
indicated.
z + 1
(i) dz (y = circle centre 1, radius 1).
;' c - 1

(ii) dz
;' . :2 + :7r2
dz
(iii) (y = circle centre 0, radius 3).
;' . z2 - 6c + 8
Prove that if f (z) is differentiable irlside and on the closed contour y, then
for any a inside y

f ( 1 flz) dz
a ) = z ;.r.j jy .z ..s

(Cauchy's integral formula). What is the value of this integral if a is outside y ?