Use Theorem 18.3.2 to evaluate the given integral. Write...

Use Theorem 18.3.2 to evaluate the given integral. Write each answer in the form $a+i b$.
$\int_{1-i}^{1+\sqrt{3 i}}\left(\frac{1}{z}+\frac{1}{z^{2}}\right) d z, C$ is any contour in the right half-plane $\operatorname{Re}(z)>0$

Key Concepts

Analytic Functions

Analytic functions are functions that are complex differentiable in a neighborhood of every point in their domain. This property ensures a wealth of powerful results in complex analysis, including integration-related theorems. When a function is analytic in a given domain, its behavior and properties are very well controlled, which is essential for both evaluating integrals and applying further theoretical results.

Contour Integration

Contour integration is the process of integrating complex functions along specified paths in the complex plane. This technique is central in complex analysis because it not only generalizes the concept of definite integrals but also allows one to use the geometric properties of the paths. The choice of contour can often simplify or generalize an evaluation, especially when the integrand is analytic in the domain of integration.

Fundamental Theorem of Calculus in Complex Analysis

The fundamental theorem of calculus for complex functions states that if a function is analytic and an antiderivative exists in a region, the integral of the function along any smooth contour between two points depends only on the values of its antiderivative at those endpoints. This principle is crucial because it provides a powerful method for evaluating complex integrals, reducing the problem to finding an antiderivative and then computing its difference at the endpoints.

Simply-Connected Domains

A simply-connected domain is a region in the complex plane that has no holes, meaning any closed contour within the domain can be continuously shrunk to a point without leaving the domain. This property guarantees that if a function is analytic in the domain, its integral over any closed contour vanishes (by Cauchy's theorem) and ensures that antiderivatives exist, which is critical for applying the fundamental theorem of calculus in complex analysis.

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