Complex Integral II

Problem 138: Evaluate the following complex integral

(1) $\begin{equation*} \int_{1}^{2} \left(\frac{1}{t}-i\right)^2 \ dt.\end{equation*}$

Solution: When evaluating complex integral, we consider the real part and imaginary part of the integral separately. That is,

(2) $\begin{align*} \int_{1}^{2} \left(\frac{1}{t}-i\right)^2 \ dt & = \int_{1}^{2} \frac{1}{t^2} - \frac{2i}{t} +i^2\ dt \\ & = \underbrace{\int_{1}^{2} \left(\frac{1}{t^2} - 1\right) \ dt}_{\text{real part}} -2i \underbrace{\int_{1}^{2} \left(\frac{1}{t}\right) \ dt}_{\text{imaginary part}} \\ & = -\frac{1}{t} - t \bigg|_{1}^{2} -2i\ln(t)\bigg|_{1}^{2} \\ & = -\frac{5}{2} - (-2) -2i\left(\ln(2) - \ln(1)\right) \\ & = -\frac{5}{2} + 2 - 2i\ln(2) \\ & = - \frac{1}{2} - i \ln(2^2) \\ & = -\frac{1}{2} - i \ln(4). \end{align*}$

That is,

(3) $\begin{equation*} \int_{1}^{2} \left(\frac{1}{t}-i\right)^2 \ dt= -\frac{1}{2} -i \ln(4).\end{equation*}$

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