Glossary

9.3 Evaluation of real integrals using contour integration

3 min read•august 7, 2024

is a powerful technique for evaluating real integrals. By extending the integration into the complex plane, we can use the to simplify calculations that would be difficult or impossible using real analysis alone.

This method is particularly useful for improper integrals and integrals involving trigonometric or rational functions. We'll explore different contour types and learn how to choose the best one for a given problem.

Contour Integration Techniques

Contour Selection and Properties

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  • Contour integration evaluates complex integrals along closed curves in the complex plane
    • Utilizes and residue theorem to simplify calculations
    • Contour choice depends on the integrand's properties (singularities, branch cuts, asymptotic behavior)
  • consists of a straight line segment and a semicircular arc
    • Useful for integrals with singularities on the real axis or at infinity
    • Radius of the semicircle is taken to infinity, allowing the use of or estimation lemmas
  • deforms around singularities or branch cuts
    • Avoids crossing branch cuts or enclosing undesired singularities
    • Indentation size is taken to zero after applying the residue theorem
  • encircles branch points or cuts in a keyhole-like shape
    • Consists of two overlapping circles connected by narrow line segments
    • Allows for the evaluation of integrals with branch points (logarithmic or fractional power singularities)

Contour Deformation and Parametrization

  • continuously modifies the integration path without crossing singularities
    • Preserves the integral's value due to Cauchy's integral theorem
    • Enables the use of more convenient contours for evaluation
  • of contours expresses the complex variable in terms of a real parameter
    • Facilitates the calculation of line integrals along the contour
    • Common parametrizations include z=reiθz = re^{i\theta} for circular arcs and z=a+tiz = a + ti for line segments (a,r,t∈Ra, r, t \in \mathbb{R})

Applications of Contour Integration

Evaluation of Real Integrals

  • Trigonometric integrals can be converted to complex contour integrals
    • Euler's formula (eix=cos⁡x+isin⁡xe^{ix} = \cos x + i \sin x) transforms trigonometric functions into complex exponentials
    • Contour integration techniques, such as residue theorem, simplify the evaluation (∫02πdθ1+cos⁡θ\int_{0}^{2\pi} \frac{d\theta}{1+\cos\theta})
  • Improper integrals with infinite limits or singularities can be evaluated using contour integration
    • Appropriate contour choice (semicircular, indented) handles the singularities or infinite limits
    • Cauchy's integral formula or residue theorem reduces the integral to a sum of residues (∫−∞∞dxx2+1\int_{-\infty}^{\infty} \frac{dx}{x^2+1})

Integral Transforms and Special Functions

  • can be evaluated using contour integration
    • Bromwich contour (a vertical line in the complex plane) is used for the inverse Fourier transform
    • Residue theorem simplifies the integral to a sum of residues at the poles of the integrand (∫−∞∞eitxx2+1dx\int_{-\infty}^{\infty} \frac{e^{itx}}{x^2+1}dx)
  • Integral representations of special functions (Gamma, Beta, Bessel) can be derived using contour integration
    • Appropriate contour choice and residue theorem lead to the integral representation
    • Helps in understanding the properties and relationships between special functions (Γ(z)=∫0∞tz−1e−tdt\Gamma(z) = \int_{0}^{\infty} t^{z-1}e^{-t}dt)

Important Theorems

Jordan's Lemma

  • Jordan's lemma estimates the contribution of a semicircular contour to an integral
    • Applies to integrals of the form ∫Cf(z)eiazdz\int_{C} f(z)e^{iaz}dz, where CC is a semicircular contour and a∈Ra \in \mathbb{R}
    • If ∣f(z)∣≤M|f(z)| \leq M on the semicircular arc and a>0a > 0, the integral over the arc vanishes as the radius tends to infinity
  • Useful in evaluating real integrals by contour integration
    • Allows the neglection of the semicircular part of the contour as the radius approaches infinity
    • Simplifies the integral to the contributions from the real axis and residues within the contour (∫−∞∞sin⁡xxdx\int_{-\infty}^{\infty} \frac{\sin x}{x}dx)

Key Terms to Review (22)

Absolute convergence: Absolute convergence refers to a condition where a series converges when the absolute values of its terms are summed. This means that if the series $$\sum a_n$$ converges absolutely, then the series $$\sum |a_n|$$ also converges. This concept is crucial because absolute convergence implies convergence of the original series, allowing for manipulation of the terms without affecting the limit.
Cauchy's Integral Theorem: Cauchy's Integral Theorem states that if a function is analytic (holomorphic) in a simply connected domain, then the integral of that function over any closed contour in that domain is zero. This fundamental result links complex analysis to contour integration, allowing for the evaluation of integrals and establishing the groundwork for other important results such as Cauchy's Integral Formula.
Complex exponential function: The complex exponential function is defined as $$e^{ix} = \cos(x) + i\sin(x)$$, where $$i$$ is the imaginary unit. This function represents a powerful bridge between exponential growth and oscillatory behavior, linking complex numbers to trigonometric functions. It is fundamental in many areas of mathematics and physics, particularly in the evaluation of integrals through techniques like contour integration, where it simplifies calculations and reveals deeper relationships in complex analysis.
Contour Deformation: Contour deformation refers to the process of altering the contour path used in complex integration to facilitate the evaluation of real integrals. This technique often involves manipulating the path of integration in the complex plane to avoid singularities and capture the contributions of residues, enabling a more straightforward computation of integrals that may be difficult or impossible to evaluate using real analysis alone.
Contour Integration: Contour integration is a method used in complex analysis to evaluate integrals along paths, or contours, in the complex plane. This technique allows for the calculation of integrals that might be difficult or impossible to evaluate using traditional real analysis methods. It connects closely with concepts like complex functions and their mappings, providing powerful tools such as the residue theorem for evaluating integrals and solving problems involving differentiation of complex functions.
Definite integral: A definite integral is a mathematical concept that represents the signed area under a curve defined by a function between two specific limits. It provides a way to calculate the accumulation of quantities, such as distance or area, over an interval and is evaluated using the Fundamental Theorem of Calculus. This tool is essential for connecting concepts in calculus with various applications, including physics and engineering.
Deformation of Contour: Deformation of contour refers to the process of changing the path of integration in complex analysis to facilitate the evaluation of integrals, particularly real integrals. This technique often involves manipulating the contour in the complex plane to simplify calculations or avoid singularities, thereby allowing for a more straightforward approach to determining the value of a given integral.
Essential Singularity: An essential singularity is a point at which a complex function exhibits behavior that is neither removable nor a pole, leading to a lack of limits or values in any neighborhood around that point. This type of singularity presents an unpredictable and infinite range of outputs, making it fundamentally different from other singularities like poles. Essential singularities are crucial in the study of analytic functions, residues, and complex integrals, as they affect the overall behavior and evaluation of functions around them.
Fourier transforms: Fourier transforms are mathematical tools that convert a function of time (or space) into a function of frequency. They provide a way to analyze the frequencies contained in a signal, making them essential in many areas of physics and engineering, particularly for solving differential equations and analyzing waveforms. By transforming real-valued functions into the frequency domain, Fourier transforms simplify the evaluation of complex integrals and enhance our understanding of periodic phenomena.
Improper integral: An improper integral is an integral that cannot be evaluated using the standard techniques of integration due to infinite limits of integration or an integrand that approaches infinity within the limits of integration. This concept is crucial for evaluating real integrals, especially when using contour integration methods to handle integrals over certain domains that include singularities or infinite intervals.
Indented Contour: An indented contour is a specific type of contour integration path that includes indentations or deviations from a straight line, typically used to navigate around singularities in the complex plane. This technique helps to evaluate real integrals by avoiding points where the integrand becomes undefined or problematic. By creating a contour that strategically bypasses these singularities, one can apply the residue theorem or other methods to compute integrals that would otherwise be difficult to solve directly.
Jordan's Lemma: Jordan's Lemma is a result in complex analysis that provides a method for evaluating certain improper integrals, particularly those involving oscillatory functions. This lemma is instrumental when using contour integration techniques to compute integrals over the real line, especially when the integrand has singularities or poles. It helps simplify the evaluation of integrals by relating them to the residues of poles enclosed within a specific contour.
Keyhole Contour: A keyhole contour is a specific type of contour used in complex analysis to evaluate certain improper integrals. It resembles a keyhole shape, consisting of a large semicircular arc that connects two straight line segments along the real axis, often avoiding singularities along the way. This contour is particularly useful when integrating functions with poles or branch cuts, allowing one to compute real integrals via residues.
Laplace Transforms: Laplace transforms are integral transforms that convert a function of time, typically denoted as $f(t)$, into a function of a complex variable, usually denoted as $s$. This powerful mathematical tool simplifies the analysis of linear time-invariant systems, making it particularly useful for solving differential equations and evaluating real integrals through contour integration, as well as for studying heat, wave, and Laplace equations in physics.
Parametrization: Parametrization refers to the process of expressing a mathematical object, such as a curve or surface, using one or more parameters that define its shape and position in space. This technique is essential for converting complex geometric forms into simpler forms that can be manipulated mathematically, particularly in the context of integrals and differential equations. By representing curves and surfaces through parameters, we can apply calculus techniques like integration more effectively.
Pole: In complex analysis, a pole is a type of singularity of a function where the function approaches infinity as the input approaches a certain value. This behavior indicates that the function cannot be defined at that point, making poles critical in understanding the analytic properties of functions, especially when working with contour integrals and residues.
Residue at infinity: The residue at infinity refers to the residue of a complex function at its point at infinity. It is an important concept in complex analysis that helps in evaluating integrals over closed contours in the complex plane, particularly when applying the residue theorem. This residue plays a critical role in determining the behavior of functions as they approach infinity, which is crucial for evaluating real integrals using contour integration methods.
Residue Theorem: The residue theorem is a powerful tool in complex analysis that allows for the evaluation of contour integrals by relating them to the residues of singularities within a closed contour. It simplifies the calculation of integrals of analytic functions by focusing on isolated singularities and their residues, making it easier to compute real integrals and analyze complex functions.
Riemann surface: A Riemann surface is a one-dimensional complex manifold that allows for the multi-valued functions of complex analysis to be treated as single-valued. It serves as a bridge between algebraic geometry and complex analysis, enabling the study of complex functions in a more structured way. Riemann surfaces help resolve issues related to branch points and cuts, which occur when dealing with functions like the square root or logarithm, allowing for a deeper understanding of complex mappings and integrals.
Semicircular contour: A semicircular contour is a specific path in the complex plane used in contour integration, where the contour forms a semicircle that connects two points on the real axis. This method is particularly useful for evaluating real integrals that extend to infinity, as it allows for the application of the residue theorem and other principles of complex analysis to simplify the computation of these integrals.
Simple pole: A simple pole is a type of singularity in complex analysis where a function goes to infinity as it approaches a specific point, and it is characterized by the fact that the function can be expressed in the form \( f(z) = \frac{g(z)}{(z - z_0)} \), where \( g(z) \) is analytic and non-zero at \( z = z_0 \). This concept plays a crucial role in understanding the behavior of functions near their singular points and is essential for evaluating certain integrals using techniques like contour integration.
Uniform Convergence: Uniform convergence refers to a type of convergence of functions where a sequence of functions converges to a limiting function uniformly over its entire domain. This means that the speed of convergence is the same across the whole range, making it possible to interchange limits with integration and differentiation under certain conditions.
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