When to use the residue at infinity

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The discussion centers on the application of the residue at infinity in contour integration, specifically for rational functions, which are defined as the ratio of two polynomials. Participants clarify that the residue at infinity can be utilized effectively when dealing with rational functions due to their behavior at infinity. The conversation emphasizes the importance of understanding the conditions under which the residue at infinity is applicable, particularly in the context of contour integration methods.

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  • Understanding of contour integration techniques
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  • Study the residue theorem in complex analysis
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Students and professionals in mathematics, particularly those focused on complex analysis, as well as anyone interested in mastering contour integration techniques and the application of residues.

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why is it that we can use the residue at infinity on the methods of contour integration example on wikipedia for the last one? we can only use the residue at infinity when it is a rational function, i.e. the ratio of two polynomials, if I'm wrong when i say this, why?
 
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does anyone know? please help
 
Can you write down the example?
 

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