0) = 9 2 . Since n(|z| = 2, 0) = 1, then |z|=2 ze3 z dz = 2πi × n(|z| = 2, 0) × Res(ze3 z ; 0) = 9πi. Exercise 3. When z = 2πki, k ∈ Z, the function (1−e−z)n is zero, then 1 (1−e−z)n has a pole at z = 2πki, k ∈ Z. Let C be any regular closed curve surrounding z = 0 and not surrounding any of the other singularities: z = 2πki, k = ±1, ±2, · · · . Then it leads to n(C, 2πki) = 1, if k = 0 0, otherwise , and C dz (1 − e−z)n = 2πi k∈Z n(C, 2πki)Res 1 (1 − e−z)n ; 2πki = 2πiRes 1 (1 − e−z)n ; 0 . Next, letting ω = 1−e−z, we have e−z = 1−ω ⇒ −e−zdz = −dω ⇒ dz = dω e−z = dω 1−ω , it implies C dz (1 − e−z)n = C∗ dω ωn(1 − ω) , where C∗ is the image under ω = 1 − e−z of C. Note that: 1. 1 ωn(1−ω) has a pole at ω = 0, 1. 2. C∗ surrounds 0 and not 1 in the ω-plane. For this, we consider the following: 5