MCS 351 ENGINEERING MATHEMATICS SOLUTION OF ...

21.
İf z = x + iy, then f(z) = Re2z = 2x is analytic.
=⇒ ∫ C
Re2zdz = 0.
14.3
1.We should use Cauchy’s integral formula to solve this question.
We can sketch C : |z − i| = 2 as follows:
∮
C
f(z)
z−z 0
dz = 2πif(z 0 )
Let g(z) = z2 −4
z 2 +4 .We know that z2 + 4 = (z − 2i)(z + 2i).
z 0 − 2i = 0 ⇒ z 0 = 2i
z 1 + 2i = 0 ⇒ z 1 = −2i
C encloses the point z 0 = 2i but don’t encloses the point z 1 = −2i, where g(z) is not analytic. Let
f(z) = z2 −4
z+2i
, D be the union of C and the interior part of C. f(z) is analytic in D. So we can use the
formula.
∮
z 2 −4
C z 2 +4 dz = ∮ z 2 −4
C z+2i . 1
z−2i dz = 2πi.f(2i) = 2πi. (2i)2 −4
2i+2i
= 2πi. 4i2 −4
4i
= 2πi. −8
4i
== −4π
4. We sketch C : |z| = π/2 as follows:
39