In mathematics or physics we are often tasked with calculating definite integrals, which by itself is a tough task, as most integrals cannot be expressed with elementary functions. We can avoid this problem if we decide to calculate the integral numerically however, if even this fails then we must find another way. One such way is with the help of the residue theorem, which we will cover in this work. We will also consider different ways to prove the convergence of definite integrals and along the way find many different types of integrals, for which we will be able to compute the value immediately. Lastly, we will analyze different ways of extending real functions to the complex plane, which will help us immensely in identifying our starting integral as either the real or the imaginary part of a complex one.
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