# Pole expansion of meromorphic functions

• LAHLH
In summary: In other words, it's saying that the function's domain is contained in the circle C_n with radius R_n.
LAHLH
Hi,

I was hoping someone may be able to help me understand what Arfken is doing in sec 7.2 when he does the pole expansion of meromorphic functions (he says this proof is due to Mittag-Leffler).

So he starts off with a function f(z) that is analytic except at some isolated poles. These poles are at isolated $z=a_n$ with $0<|a_1|<|a_2|<...$ and are all simple with residues $b_n$. He then considers a series of concentric circles C_n about the origin so that C_n contains poles $a_1,a_2,...a_n$ but no others. Finally he assumes that $|f(z)|<\epsilon R_n$ where R_n is radius of C_n and $\epsilon>0$ is small constant. He says then that:

$$f(z)=f(0)+\sum_0^{\infty} b_n\{(z-a_n)^{-1}+a_n^{-1}\}$$

converges to f(z).

To prove this he says to prove this we use residue theorem to evaluate the contour integral for z inside C_n:

$$I_n:=(2\pi i)^{-1}\int_{C_n}\,\frac{f(w)}{w(w-z)}\mathrm{d}w$$

$$I_n=\sum_{m=1}^{n}\frac{b_m}{a_m(a_m-z)}+\frac{f(z)-f(0)}{z}$$

My first question is how to prove this second equality using the residue theorem? I just don't seem to be able to get it out..

Suppose we have two functions f and g that are holomorphic inside some circle ##\gamma## except for poles a_1, ..., a_n for f and b_1, ..., b_m for g. Also suppose that no a_i is equal to a b_j. Then the residue theorem yields
$$(2\pi i)^{-1} \int_\gamma f(z)g(z) dz = \sum_{i=1}^n \text{Res}(f,a_i)g(a_i) + \sum_{j=1}^m \text{Res}(g,b_j)f(b_j).$$

In our case g(w)=1/w(w-z), which has poles at 0 and z with residues -1/z and 1/z, resp. So
\begin{align} (2\pi i)^{-1} \int_{C_n} \frac{f(w)}{w(w-z)} dw &= (2\pi i)^{-1} \int_{C_n} f(w) g(w) dw \\ &= \sum_{m=1}^n b_m g(a_m) + \text{Res}(g,0)f(0) + \text{Res}(g,z)f(z) & \\ &= \sum_{m=1}^n \frac{b_m}{a_m(a_m-z)} + \frac{f(z)-f(0)}{z}, \end{align}
as desired.

[Note that this only holds if ##a_m,z\neq0## and ##z\neq a_m## for all m; this is consistent with the requirement that the poles of f and g be distinct.]

morphism said:
Suppose we have two functions f and g that are holomorphic inside some circle ##\gamma## except for poles a_1, ..., a_n for f and b_1, ..., b_m for g. Also suppose that no a_i is equal to a b_j. Then the residue theorem yields
$$(2\pi i)^{-1} \int_\gamma f(z)g(z) dz = \sum_{i=1}^n \text{Res}(f,a_i)g(a_i) + \sum_{j=1}^m \text{Res}(g,b_j)f(b_j).$$

In our case g(w)=1/w(w-z), which has poles at 0 and z with residues -1/z and 1/z, resp. So
\begin{align} (2\pi i)^{-1} \int_{C_n} \frac{f(w)}{w(w-z)} dw &= (2\pi i)^{-1} \int_{C_n} f(w) g(w) dw \\ &= \sum_{m=1}^n b_m g(a_m) + \text{Res}(g,0)f(0) + \text{Res}(g,z)f(z) & \\ &= \sum_{m=1}^n \frac{b_m}{a_m(a_m-z)} + \frac{f(z)-f(0)}{z}, \end{align}
as desired.

[Note that this only holds if ##a_m,z\neq0## and ##z\neq a_m## for all m; this is consistent with the requirement that the poles of f and g be distinct.]

Ah, thank you very much, very helpful!

"he assumes that |f(z)|<ϵRn where R_n is radius of C_n and ϵ>0 is small constant. "

how can a function with poles be bounded?

Presumably that inequality is for points z on C_n.

## 1. What is the "Pole expansion" of a meromorphic function?

The "Pole expansion" of a meromorphic function is a mathematical technique used to represent a meromorphic function as a sum of simpler functions. It allows us to understand the behavior of the function near its poles, which are points where the function is undefined or infinite.

## 2. Why is the "Pole expansion" important in the study of meromorphic functions?

The "Pole expansion" is important because it allows us to analyze the behavior of meromorphic functions near their poles, which are points where the function has singularities. This helps us understand the overall behavior of the function and make predictions about its properties.

## 3. How is the "Pole expansion" different from the Taylor series expansion?

While the Taylor series expansion is used to represent a function as a sum of powers of the variable, the "Pole expansion" focuses on representing a meromorphic function as a sum of simpler functions near its poles. The "Pole expansion" is typically used for functions with singularities, while the Taylor series is used for smooth functions.

## 4. Can the "Pole expansion" be used for functions with an infinite number of poles?

Yes, the "Pole expansion" can be used for functions with an infinite number of poles. In this case, the expansion will involve an infinite sum, and its convergence will depend on the properties of the function and its poles.

## 5. How is the "Pole expansion" related to the Residue Theorem?

The "Pole expansion" is closely related to the Residue Theorem in complex analysis. In fact, the Residue Theorem can be used to calculate the coefficients in the "Pole expansion" of a meromorphic function. This allows us to easily find the poles and their corresponding residues, which are important in understanding the behavior of the function near its poles.

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