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**Unformatted text preview: **Z ∞ sin x x dx Hint: Integrate e iz /z around the contour below. (Prove any limits you use). C R C ε ε R . 6. Suppose that f is an analytic function on the complex plane minus the two points ± 1. Let γ 1 denote the curve given by z 1 ( t ) = 1 + e it where 0 ≤ t ≤ 2 π and let γ 2 denote the curve given by z 2 ( t ) =-1 + e it where 0 ≤ t ≤ 2 π . Suppose that Z γ j f ( z ) dz = 0 for j = 1 , 2. First, explain why Z γ f ( z ) dz = 0 for any closed curve in C-{± 1 } . Next, prove from ﬁrst principles that f has an analytic antiderivative on C- {± 1 } , i.e., show that there is an analytic function g on C-{± 1 } such that g = f ....

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