In complex analysis, the residue at infinity is an important concept used in the evaluation of contour integrals, particularly in the context of the residue theorem. Here’s a summary of the concept and how it works:

Definition of Residue at Infinity

If $f(z)$ is a meromorphic function on the extended complex plane $\mathbb{C} \cup \{\infty\}$, then the residue of $f$ at infinity, denoted $\operatorname{Res}(f, \infty)$, is defined by the property that the sum of the residues of $f$ at all of its finite poles plus the residue at infinity equals zero.

Mathematically, $\operatorname{Res}(f, \infty) = -\sum_{z \in \mathbb{C}} \operatorname{Res}(f, z)$ where the sum runs over all finite poles $z$ of $f(z)$.

How to Calculate the Residue at Infinity

To find the residue at infinity, you can use the following steps:

1. Change of Variable: Rewrite the function $f(z)$ in terms of the variable $w = \frac{1}{z}$. Then $z = \frac{1}{w}$ and $dz = -\frac{1}{w^2} \, dw$.

2. Express $f(z)$ in Terms of $w$: Substitute $z = \frac{1}{w}$ into $f(z)$ to get a new function $g(w) = f\left(\frac{1}{w}\right)$.

3. Find the Residue of $-g(w)$ at $w = 0$: The residue at infinity of $f(z)$ is then defined as $-\operatorname{Res}(g(w), 0)$.

Example

For example, consider $f(z) = \frac{1}{z^2 + 1}$. The poles of this function are at $z = i$ and $z = -i$. Calculating the residues at $z = i$ and $z = -i$, we can use the above relationship to find the residue at infinity: $\operatorname{Res}(f, \infty) = -\left( \operatorname{Res}(f, i) + \operatorname{Res}(f, -i) \right)$

Would you like more details on this calculation, or perhaps examples of specific functions?

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1. What are some practical applications of finding residues at infinity in physics or engineering?
2. How do we determine the order of a pole when calculating residues?
3. How does the residue at infinity relate to Laurent series expansions?
4. What is the geometric interpretation of the residue at infinity?
5. Could you apply the residue theorem to compute integrals over closed contours?

Tip: When calculating residues, remember that simple poles are often easier to handle, while higher-order poles might require differentiation.