6.1 Residues

Residues are complex numbers tied to isolated singularities of complex functions. They're key in evaluating complex integrals using the residue theorem, which relates contour integrals to the sum of residues at poles inside the contour.

This powerful tool simplifies calculations for otherwise tricky integrals. By identifying poles and their residues, we can solve complex problems in various fields, from fluid dynamics to quantum mechanics, without directly evaluating the integral.

Definition of residues

• Residues are complex numbers associated with isolated singularities of a complex function
• Play a crucial role in evaluating complex integrals using the residue theorem
• Provide a way to calculate the value of a contour integral without explicitly evaluating the integral itself

Residue theorem

Contour integrals and residues

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• The residue theorem relates the value of a contour integral of a meromorphic function to the sum of the residues at its poles inside the contour
• Enables the calculation of complex integrals by identifying the poles and their corresponding residues
• Simplifies the evaluation of integrals that would otherwise be difficult or impossible to compute directly

Poles and residues

• Poles are isolated singularities of a complex function where the function becomes undefined or goes to infinity
• Each pole has an associated residue, which quantifies the behavior of the function near the pole
• The residue at a pole determines the contribution of that pole to the contour integral

Simple poles

• A simple pole is a pole of order one, where the function has a single, non-zero residue
• For a simple pole at $z = z_0$, the residue is given by $\lim_{z \to z_0} (z - z_0) f(z)$
• Simple poles are the most common type of poles encountered in complex analysis

Poles of higher order

• A pole of order $n > 1$ is a pole where the function has a zero of order $n$ in the denominator
• The residue at a pole of order $n$ can be calculated using the limit formula or by expanding the function into a Laurent series
• Higher-order poles contribute to the contour integral based on their residues and the order of the pole

Calculating residues

Limit definition

• The residue at a pole $z = z_0$ can be calculated using the limit formula: $\text{Res}[f(z), z_0] = \frac{1}{(n-1)!} \lim_{z \to z_0} \frac{d^{n-1}}{dz^{n-1}} [(z - z_0)^n f(z)]$
• For simple poles ($n = 1$), the formula simplifies to $\lim_{z \to z_0} (z - z_0) f(z)$
• The limit definition is a direct way to calculate residues, especially for simple poles

Power series expansion

• Residues can be calculated by expanding the function into a power series around the pole
• The coefficient of the $(z - z_0)^{-1}$ term in the power series expansion gives the residue at $z = z_0$
• Power series expansion is useful when the function can be easily expressed as a power series

Laurent series

• The Laurent series is a generalization of the power series that allows for negative powers of $(z - z_0)$
• The residue at a pole $z = z_0$ is the coefficient of the $(z - z_0)^{-1}$ term in the Laurent series expansion
• Laurent series expansion is a powerful tool for calculating residues, especially for higher-order poles

Applications of residues

Evaluating real integrals

• Residue theorem can be used to evaluate real integrals by extending the integrand to the complex plane and choosing an appropriate contour
• The integral is then expressed as a sum of residues, which can be easier to calculate than the original integral
• Examples include integrals of rational functions, trigonometric functions, and logarithmic functions

Improper integrals

• Residue theorem is particularly useful for evaluating improper integrals, where the integrand has singularities or the limits of integration are infinite
• By choosing a suitable contour and applying the residue theorem, the improper integral can be transformed into a sum of residues
• Examples include integrals with poles on the real axis or at infinity

Definite integrals

• Definite integrals can be evaluated using the residue theorem by extending the integrand to the complex plane and choosing a closed contour that includes the interval of integration
• The integral is then expressed as a sum of residues and contributions from the contour at infinity
• This method is especially useful for integrals involving trigonometric or exponential functions

Infinite series

• Residue theorem can be used to evaluate infinite series by expressing the series as a contour integral and applying the residue theorem
• The sum of the series is then related to the residues of the associated complex function
• Examples include summing series involving rational functions or trigonometric functions

Laplace transforms

• Residue theorem is used in the inversion of Laplace transforms, which is the process of recovering the original function from its Laplace transform
• The inverse Laplace transform can be expressed as a contour integral, and the residue theorem is applied to evaluate the integral
• This method is particularly useful for finding the inverse Laplace transform of rational functions

Examples of residue calculations

Simple pole examples

• $f(z) = \frac{1}{z - 1}$, pole at $z = 1$, residue = 1
• $f(z) = \frac{z}{(z - 2)(z + 3)}$, poles at $z = 2$ and $z = -3$, residues = $\frac{2}{5}$ and $-\frac{3}{5}$

Higher order pole examples

• $f(z) = \frac{1}{(z - 1)^2}$, pole of order 2 at $z = 1$, residue = 1
• $f(z) = \frac{z^2}{(z - 2)^3}$, pole of order 3 at $z = 2$, residue = 4

Multiple pole examples

• $f(z) = \frac{1}{(z - 1)(z - 2)^2}$, simple pole at $z = 1$ and pole of order 2 at $z = 2$, residues = $\frac{1}{3}$ and $-\frac{1}{3}$
• $f(z) = \frac{z^2}{(z - 1)^2(z + 1)^3}$, poles of order 2 at $z = 1$ and order 3 at $z = -1$, residues = $-\frac{1}{4}$ and $\frac{1}{8}$

Residues and branch cuts

Logarithmic branch points

• Logarithmic functions have branch points where the function is multi-valued
• When integrating functions with logarithmic branch points, the choice of branch cut affects the value of the integral
• The residue theorem can be applied by choosing an appropriate branch cut and accounting for the contributions from the branch cut

Square root branch points

• Square root functions have branch points where the function is multi-valued
• Similar to logarithmic branch points, the choice of branch cut affects the value of the integral when applying the residue theorem
• The contributions from the branch cut must be considered along with the residues when evaluating the integral

Residue theorem vs Cauchy's integral formula

• Cauchy's integral formula relates the value of a holomorphic function at a point to a contour integral of the function
• The residue theorem is a generalization of Cauchy's integral formula for meromorphic functions, which may have poles
• While Cauchy's integral formula is used for functions without poles, the residue theorem is used for functions with poles, relating the contour integral to the sum of residues

Residues in physical applications

Fluid dynamics

• Residue theorem is used in the analysis of complex potential flows, such as in aerodynamics and hydrodynamics
• The complex potential function describes the velocity field of the fluid, and its singularities correspond to sources, sinks, or vortices
• The residue theorem is applied to calculate the circulation around vortices and to determine the lift force on airfoils

Electromagnetism

• Residue theorem is used in the calculation of electromagnetic fields and potentials in the presence of sources or singularities
• The complex potential functions in electrostatics and magnetostatics can be analyzed using the residue theorem
• Examples include calculating the electric field of a point charge or the magnetic field of a current loop

Quantum mechanics

• Residue theorem is used in the evaluation of integrals that arise in quantum mechanics, such as in the calculation of transition amplitudes or Green's functions
• The poles of the integrand correspond to the energy levels of the quantum system, and the residues determine the contributions of each energy level
• The residue theorem simplifies the calculation of these integrals, which are essential for understanding the behavior of quantum systems