5. Prove Corollary 5.3.4 directly from Theorem 4.3.3. Do not appeal to the residue theorem. 6. Let Ω ⊆ C be open such that −z ∈ Ω whenever z ∈ Ω. Suppose f is even and analytic on Ω except for...

5. Prove Corollary 5.3.4 directly from Theorem 4.3.3. Do not appeal to the residue theorem.

6. Let Ω
⊆
C be open such that
−
z

∈
Ω whenever
z

∈
Ω. Suppose
f
is even and analytic on Ω except for isolated singularities in Ω. Prove that if
a

∈
Ω is a pole of order
m

∈
N or an essential singularity of
f
, then
−
a
is a singularity of the same type and

Res
f
(z) =
−
Res
f
(z).

z=−
a

z=a

f
(z)
dz
= 0?