Abstract:

Residue theorem has been frequently used to tackle certain complicated deffinite integrals. However, it is never applied for indeffinite integrals. Therefore, in this report, residue theorem and a some small tricks are applied to find antiderivatives. The are mainly three interesting results:

1. Antiderivatives can be found without integration: antiderivatives can be represented by residues, while calculation of residues requires no knowledge of integration. For residues at poles, only differentiation is needed.

For residues at essential singularities, Taylor series manipulation is required; still, it is just differentiation with algebra work. This allows fast computation of antiderivatives of rational functions, especially those with only simple poles, providing an alternative to partial fraction decomposition. This is also applicable for other functions. Moreover, this result implies that integration is not only the reverse of differentiation, integration is indeed equivalent to differentiation.

2. A universal functional form of antiderivative can be obtained: antiderivatives obtained by this method has a functional form that converges wherever it should converge. The functional form has the largest possible region of convergence on the complex plane.

3. As a weak tool for analytic continuation: since the universal functional form of antiderivative is obtained, differentiating yields a universal functional form of the integrand. If one knows the behaviour of f in the vicinity of every singularity of f, one can analytically continue f to its largest possible domain by the method presented in this report.

Residue theorem is the central tool to be used throughout this report. Certain simple inequalities such as triangle inequality and estimation lemma are occasionally applied

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