10.2 Meromorphic functions

Meromorphic functions are like complex-valued functions with a few hiccups. They're smooth sailing everywhere except for some isolated points called poles. These functions are crucial in complex analysis, helping us understand how functions behave in the complex plane.

In this part, we'll dive into the nitty-gritty of meromorphic functions. We'll look at their poles, residues, and how they behave at infinity. Plus, we'll see how the residue theorem can help us solve tricky integrals. It's all about understanding these functions' quirks and using them to our advantage.

Meromorphic functions and their domain

Definition and properties

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• A meromorphic function is a function that is holomorphic on a domain except for a set of isolated points, which are poles of the function
• Meromorphic functions can be expressed as the ratio of two holomorphic functions, where the denominator has isolated zeros
• The set of poles of a meromorphic function is a discrete subset of the domain
• Meromorphic functions are analytic on their domain except at the poles

Domain and singularities

• The domain of a meromorphic function is the set of all points in the complex plane where the function is defined, excluding the poles
• Poles are isolated singularities where the function is undefined but can be made defined by multiplying the function by a suitable power of $(z - z₀)$, where $z₀$ is the pole
• Examples of meromorphic functions include rational functions ($f(z) = \frac{P(z)}{Q(z)}$, where $P(z)$ and $Q(z)$ are polynomials) and trigonometric functions ($\tan(z)$, $\csc(z)$)

Poles and residues of meromorphic functions

Poles and their order

• A pole of a meromorphic function is a point where the function is undefined but can be made defined by multiplying the function by a suitable power of $(z - z₀)$, where $z₀$ is the pole
• The order of a pole is the smallest positive integer $n$ such that $(z - z₀)ⁿf(z)$ has a finite, non-zero limit as $z$ approaches $z₀$
• Simple poles have order 1, while higher-order poles have order greater than 1
• Examples of functions with poles include $f(z) = \frac{1}{z^2 + 1}$ (simple poles at $z = \pm i$) and $g(z) = \frac{1}{(z - 1)^3}$ (pole of order 3 at $z = 1$)

Residues and their calculation

• The residue of a meromorphic function at a pole is the coefficient of the $(z - z₀)⁻¹$ term in the Laurent series expansion of the function around the pole
• For a simple pole (order 1), the residue can be calculated using the limit: $Res(f, z₀) = \lim_{z \to z₀} (z - z₀)f(z)$
• For higher-order poles, the residue can be found using the formula: $Res(f, z₀) = \frac{1}{(n-1)!} \lim_{z \to z₀} \frac{d^{n-1}}{dz^{n-1}} [(z - z₀)ⁿf(z)]$, where $n$ is the order of the pole
• Examples of residue calculations include $f(z) = \frac{1}{z^2 + 1}$ (residues $\frac{i}{2}$ at $z = i$ and $-\frac{i}{2}$ at $z = -i$) and $g(z) = \frac{1}{(z - 1)^3}$ (residue $\frac{1}{2}$ at $z = 1$)

Residue theorem for integral evaluation

Statement and application of the theorem

• The residue theorem states that the integral of a meromorphic function $f(z)$ along a closed contour $C$ is equal to $2πi$ times the sum of the residues of $f(z)$ at the poles enclosed by $C$
• To apply the residue theorem, identify the poles of the meromorphic function within the contour and calculate their residues
• If a pole lies on the contour, the contour should be deformed to exclude the pole or include it completely, depending on the problem's requirements

Examples and techniques

• When evaluating real integrals using the residue theorem, choose a suitable contour (e.g., a semicircle) and use the theorem to simplify the integral
• The residue theorem can be used to evaluate integrals of rational functions, trigonometric functions, and other meromorphic functions over closed contours
• Examples of integrals that can be evaluated using the residue theorem include $\int_{-\infty}^{\infty} \frac{dx}{x^2 + 1}$ (using a semicircular contour in the upper half-plane) and $\int_{0}^{2\pi} \frac{d\theta}{a + b\cos\theta}$ (using the unit circle as the contour)

Meromorphic functions at infinity

Behavior at infinity

• The behavior of a meromorphic function at infinity can be studied by considering the function $f(1/z)$ and examining its behavior near $z = 0$
• A meromorphic function has a pole at infinity if $\lim_{z \to \infty} f(z) = \infty$, and the order of the pole is the order of the zero of $f(1/z)$ at $z = 0$
• A meromorphic function has a removable singularity at infinity if $\lim_{z \to \infty} f(z)$ exists and is finite, and $f(1/z)$ has a removable singularity at $z = 0$
• A meromorphic function has an essential singularity at infinity if $\lim_{z \to \infty} f(z)$ does not exist, and $f(1/z)$ has an essential singularity at $z = 0$

Residue at infinity and its applications

• The residue at infinity can be calculated using the residue at $z = 0$ for the function $f(1/z)$ multiplied by $-1$
• The behavior of a meromorphic function at infinity is crucial when applying the residue theorem over unbounded contours or when studying the global properties of the function
• Examples of functions with poles at infinity include $f(z) = z^2$ (pole of order 2 at infinity) and $g(z) = \frac{1}{z}$ (simple pole at infinity)
• The residue at infinity can be used to evaluate integrals over unbounded contours, such as $\int_{-\infty}^{\infty} \frac{dx}{x^2 + 1}$ (using a semicircular contour in the upper half-plane and the residue at infinity)