How to calculate residue at infinity formula and examples.
#residue #infinity #residuearinfinity #examples
The residue at infinity is a concept in complex analysis that helps us understand how functions behave near infinity. It is useful for studying the growth and singularities of functions as they approach infinity.
To find the residue at infinity, we consider a complex function f(z) defined on the extended complex plane. We represent the point at infinity as z = 1/w, where w is a finite complex number. By rewriting the function in terms of w, we get a new function g(w), usually a fraction. The residue at infinity, denoted Res[g(w), w = 0], is the coefficient of the highest power of w in the Laurent series expansion of g(w) around w = 0.
The residue at infinity tells us how the function behaves as z approaches infinity. A residue of zero means the function behaves nicely, while a non-zero residue indicates more complex behavior.
The residue at infinity is important in studying meromorphic functions and contour integrals. By calculating the residues at finite points and the residue at infinity, we can use the residue theorem to evaluate these integrals.
In summary, the residue at infinity helps us understand how functions behave near infinity. It provides insights into growth, singularities, and integral properties of functions on the extended complex plane.
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