Residue at multiple order poles is a mathematical concept related to complex analysis functions that have poles, which are points where the function becomes infinite. The residue of a function at a pole provides information about the behavior of the function near that pole. The order of a pole refers to the number of times the function becomes infinite at that point. Understanding the concepts of order, singularity, Laurent expansion, and contour integration is crucial for evaluating residues at multiple order poles.
Structure for Residue at Multiple Order Poles
When a function has a pole of order $m$ at $z=a$, its residue is given by the formula
$$\text{Res}(f,z=a) = \lim_{z\to a} \frac{1}{(m-1)!} \frac{d^{m-1}}{dz^{m-1}} [(z-a)^m f(z)]$$
If the function has multiple order poles at the same point, the residue is the sum of the residues at each pole.
For example, if the function $f(z)$ has a pole of order $2$ at $z=0$ and a pole of order $3$ at $z=1$, then the residue of $f(z)$ at $z=0$ is given by
$$\text{Res}(f,z=0) = \lim_{z\to 0} \frac{1}{(2-1)!} \frac{d^{2-1}}{dz^{2-1}} [(z-0)^2 f(z)] = \lim_{z\to 0} \frac{1}{1} \frac{d}{dz} [z^2 f(z)] = \lim_{z\to 0} z^2 f'(z)$$
and the residue of $f(z)$ at $z=1$ is given by
$$\text{Res}(f,z=1) = \lim_{z\to 1} \frac{1}{(3-1)!} \frac{d^{3-1}}{dz^{3-1}} [(z-1)^3 f(z)] = \lim_{z\to 1} \frac{1}{2} \frac{d^2}{dz^2} [(z-1)^3 f(z)]$$
The total residue of $f(z)$ at $z=0$ and $z=1$ is therefore
$$\text{Res}(f,z=0) + \text{Res}(f,z=1) = \lim_{z\to 0} z^2 f'(z) + \lim_{z\to 1} \frac{1}{2} \frac{d^2}{dz^2} [(z-1)^3 f(z)]$$
In general, if the function $f(z)$ has multiple order poles at $z=a_1, a_2, \ldots, a_n$, then the residue of $f(z)$ at $z=a_i$ is given by
$$\text{Res}(f,z=a_i) = \lim_{z\to a_i} \frac{1}{(m_i-1)!} \frac{d^{m_i-1}}{dz^{m_i-1}} [(z-a_i)^{m_i} f(z)]$$
where $m_i$ is the order of the pole at $z=a_i$.
The total residue of $f(z)$ at all of its poles is therefore
$$\text{Res}(f,z=a_1) + \text{Res}(f,z=a_2) + \cdots + \text{Res}(f,z=a_n)$$
The following table summarizes the structure for residue at multiple order poles:
| Pole | Order | Residue |
|---|---|---|
| $a_1$ | $m_1$ | $\lim_{z\to a_1} \frac{1}{(m_1-1)!} \frac{d^{m_1-1}}{dz^{m_1-1}} [(z-a_1)^{m_1} f(z)]$ |
| $a_2$ | $m_2$ | $\lim_{z\to a_2} \frac{1}{(m_2-1)!} \frac{d^{m_2-1}}{dz^{m_2-1}} [(z-a_2)^{m_2} f(z)]$ |
| $\vdots$ | $\vdots$ | $\vdots$ |
| $a_n$ | $m_n$ | $\lim_{z\to a_n} \frac{1}{(m_n-1)!} \frac{d^{m_n-1}}{dz^{m_n-1}} [(z-a_n)^{m_n} f(z)]$ |
Total residue: $\text{Res}(f,z=a_1) + \text{Res}(f,z=a_2) + \cdots + \text{Res}(f,z=a_n)$
Question 1:
What is the significance of residue at multiple order poles?
Answer:
Residue at multiple order poles provides essential information about the behavior of a function in the vicinity of those poles. It determines the rate at which the function approaches infinity as the independent variable approaches the pole. The order of the pole signifies the strength of the singularity, with higher-order poles indicating a more rapid approach to infinity.
Question 2:
How is residue calculated for multiple order poles?
Answer:
For multiple order poles, the residue is calculated as the coefficient of (z – a)^-n in the Laurent series expansion of the function f(z) about the pole a, where n is the order of the pole. This coefficient can be determined using appropriate techniques such as L’Hospital’s rule or complex integration.
Question 3:
What are the applications of residue theorem for multiple order poles?
Answer:
The residue theorem for multiple order poles is widely used in various fields of mathematics and physics. It allows for the evaluation of definite integrals involving functions with multiple order poles, the calculation of sums of infinite series, and the solution of systems of linear equations with complex coefficients.
Well, my dear math enthusiasts, I hope you’ve enjoyed this excursion into the realm of residue at multiple order poles. I understand that these concepts can be a bit daunting, but I’m confident that you’ve gained a solid foundation for tackling more advanced topics in complex analysis. As always, feel free to reach out if you have any questions or need further clarification. And don’t forget to visit again soon for more mathematical adventures. Until then, keep exploring, keep questioning, and keep enjoying the beauty of the mathematical world!