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Unformatted text preview: is analytic in the disk | z | ≤ 2 since its only singularity, at z = 3, lies outside the contour C . Therefore, by Cauchy’s formula (7.135), we immediately obtain the integral contintegraldisplay C e z dz z 2 − 2 z − 3 = contintegraldisplay C f ( z ) z + 1 dz = 2 π i f ( − 1) = − π i 2 e . Note : Path independence implies that the integral has the same value on any other simple closed contour, provided it is oriented in the usual counter-clockwise direction and encircles the point z = 1 but not the point z = 3. Derivatives by Integration The fact that we can recover values of complex functions by integration is noteworthy. Even more amazing † is the fact that we can compute derivatives of complex functions by integration — turning the Fundamental Theorem on its head! Let us differentiate both sides of Cauchy’s formula (7.135) with respect to a . The integrand in the Cauchy formula is sufficiently nice so as to allow us to bring the derivative inside the integral sign. Moreover,sufficiently nice so as to allow us to bring the derivative inside the integral sign....
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