Glossary

7.2 Evaluating real integrals using residues

3 min read•august 13, 2024

is a powerful tool for evaluating tricky real integrals. By converting real integrals to complex ones, we can use and the to solve problems that would be difficult or impossible with traditional methods.

This section focuses on applying residue theory to different types of real integrals. We'll learn how to choose the right contours, convert , and handle with singularities. It's a game-changer for tackling tough integration problems.

Real integrals using residues

Types of real integrals evaluated using residues

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Top images from around the web for Types of real integrals evaluated using residues

  • complex analysis - Evaluating series by contour integration, the residue theorem, and cotangent ... View original

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  • complex analysis - Integrating $\int_0^\infty \frac{1-\cos x }{x^2}dx$ via contour integral ... View original

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  • Evaluate real integrals of the form ∫−∞∞R(x)dx\int_{-\infty}^{\infty} R(x) dx, where R(x)R(x) is a , using residues when certain conditions are met
  • Calculate real integrals of the form ∫0∞R(x)dx\int_{0}^{\infty} R(x) dx or ∫−∞0R(x)dx\int_{-\infty}^{0} R(x) dx using residues under specific conditions
  • Convert trigonometric integrals of the form ∫−ππf(cos⁡θ,sin⁡θ)dθ\int_{-\pi}^{\pi} f(\cos \theta, \sin \theta) d\theta into complex integrals and evaluate using residues
  • Evaluate improper real integrals with singularities using residues by choosing an appropriate contour (semicircular, quarter-circular)

Conditions for applying residue theory to real integrals

  • Rational function R(x)R(x) must satisfy certain conditions for residue theory to be applicable
    • Degree of denominator must be at least 2 higher than the degree of the numerator
    • Rational function must have no poles on the real axis
  • Trigonometric integrals must be convertible into complex integrals using substitution (z=eiθz = e^{i\theta})
  • Improper real integrals must have singularities that can be enclosed by an appropriate contour in the complex plane

Converting real to complex integrals

Replacing real variables with complex variables

  • Convert real integrals into complex integrals by replacing the real variable with a complex variable (x→zx \rightarrow z)
  • Choose an appropriate contour in the complex plane based on the type of real integral
    • For integrals of the form ∫−∞∞R(x)dx\int_{-\infty}^{\infty} R(x) dx, use a semicircular contour in the upper or lower half-plane, depending on the poles of R(z)R(z)
    • For integrals of the form ∫0∞R(x)dx\int_{0}^{\infty} R(x) dx or ∫−∞0R(x)dx\int_{-\infty}^{0} R(x) dx, use a quarter-circular contour in the appropriate quadrant of the complex plane

Choosing appropriate contours for trigonometric and improper integrals

  • Convert trigonometric integrals into complex integrals using the substitution z=eiθz = e^{i\theta} and a unit circle contour
    • Rewrite trigonometric functions in terms of complex exponentials (cos⁡θ=eiθ+e−iθ2\cos \theta = \frac{e^{i\theta} + e^{-i\theta}}{2}, sin⁡θ=eiθ−e−iθ2i\sin \theta = \frac{e^{i\theta} - e^{-i\theta}}{2i})
  • Choose a contour that encloses all the singularities of the integrand in the complex plane for improper real integrals
    • Singularities can include poles, branch points, or essential singularities
    • Contour choice depends on the location and type of singularities

Evaluating real integrals with residues

Applying the residue theorem

  • Use the residue theorem to evaluate complex integrals: ∮Cf(z)dz=2πi∑Res[f(z),zk]\oint_{C} f(z) dz = 2\pi i \sum \text{Res}[f(z), z_{k}], where CC is a simple closed contour, and zkz_{k} are the poles of f(z)f(z) enclosed by CC
  • Identify the poles of the complex integrand and calculate their residues
    • For simple poles, use the formula: Res[f(z),zk]=lim⁡z→zk(z−zk)f(z)\text{Res}[f(z), z_{k}] = \lim_{z \to z_{k}} (z - z_{k})f(z)
    • For poles of order nn, use the formula: Res[f(z),zk]=1(n−1)!lim⁡z→zkdn−1dzn−1[(z−zk)nf(z)]\text{Res}[f(z), z_{k}] = \frac{1}{(n-1)!} \lim_{z \to z_{k}} \frac{d^{n-1}}{dz^{n-1}} [(z - z_{k})^{n} f(z)]

Calculating the value of the real integral

  • Sum the residues multiplied by 2πi2\pi i to obtain the value of the original real integral
  • Ensure that the chosen contour encloses all the relevant poles
  • Verify that the integrand decays sufficiently quickly along the contour for the integral to converge

Interpreting results from residue theory

Analyzing the complex result

  • Interpret the result of a real integral evaluated using residues, which is a complex number
    • For a well-defined real integral, the imaginary part should be zero
    • A non-zero imaginary part may indicate that the original integral is not convergent or that the chosen contour is not appropriate

Principal value integrals and convergence

  • In some cases, interpret the real part of the result as a integral, which assigns a finite value to an otherwise divergent integral
  • Use the residue theorem to prove the convergence or divergence of certain improper real integrals by analyzing the behavior of the integrand at its singularities
    • If the integrand decays sufficiently quickly at infinity, the integral converges
    • If the integrand grows or oscillates at infinity, the integral may diverge

Key Terms to Review (20)

Cauchy Integral Formula for Derivatives: The Cauchy Integral Formula for Derivatives provides a way to calculate the derivatives of holomorphic functions within a closed contour in the complex plane. It states that if a function is holomorphic inside and on some simple closed contour C, then the n-th derivative at a point inside C can be expressed as an integral of the function over C. This formula is fundamental because it links complex integration with differentiation, showcasing the powerful relationship between these operations in complex analysis.
Cauchy's Residue Theorem: Cauchy's Residue Theorem is a fundamental result in complex analysis that provides a method for evaluating certain types of integrals by relating them to the residues of singularities within a contour. The theorem states that if a function is analytic on and inside a closed contour except for a finite number of isolated singularities, the integral of the function around the contour is equal to $2\pi i$ times the sum of the residues at those singularities. This powerful tool not only simplifies the evaluation of integrals but also connects deeply with the properties of holomorphic functions.
Contour integration: Contour integration is a technique in complex analysis that involves integrating complex functions along a specified path, or contour, in the complex plane. This method allows for the evaluation of integrals that are often difficult or impossible to compute using traditional real analysis methods, making it essential for deriving results related to residues, meromorphic functions, and various applications in physics and engineering.
Essential Singularity: An essential singularity is a type of singular point of a complex function where the behavior of the function is particularly wild and unpredictable. Unlike removable singularities or poles, an essential singularity causes the function to exhibit infinite oscillations or diverging values as it approaches that point, making it crucial in understanding the nature of complex functions and their series expansions.
Holomorphic Function: A holomorphic function is a complex function that is differentiable at every point in its domain, which also implies that it is continuous. This differentiability means the function can be represented by a power series around any point within its domain, showcasing its smooth nature. Holomorphic functions possess various important properties, including satisfying Cauchy-Riemann equations, which connect real and imaginary parts of the function and link them to complex analysis concepts like contour integrals and Cauchy's integral theorem.
Improper integrals: Improper integrals are integrals that involve either infinite limits of integration or integrands that become infinite within the limits of integration. These integrals require special techniques to evaluate because they may not converge in the traditional sense. Understanding improper integrals is essential for using advanced methods like the residue theorem and evaluating real integrals, as these tools often deal with singularities and unbounded regions.
Jordan's Lemma: Jordan's Lemma is a result in complex analysis that provides conditions under which certain integrals can be evaluated using contour integration. It is particularly useful when dealing with integrals involving oscillatory functions, as it helps to show that the contributions from certain parts of the contour vanish. This lemma plays a crucial role in simplifying the evaluation of integrals, especially when applying the residue theorem or working with semicircular contours in the upper or lower half-plane.
Laurent series expansion: The Laurent series expansion is a representation of a complex function as a power series that includes both positive and negative powers, typically used for functions that have singularities. It generalizes the Taylor series, allowing for the analysis of functions in regions that contain poles or essential singularities. This makes it particularly useful in complex analysis for evaluating integrals and understanding the behavior of functions around points where they are not analytic.
Limit of a contour integral: The limit of a contour integral refers to the evaluation of an integral along a specified path in the complex plane as certain parameters approach a particular value. This concept is crucial when determining the behavior of integrals around singularities and when applying the residue theorem to evaluate real integrals, as it helps establish the conditions under which integrals converge to a meaningful result.
Meromorphic Function: A meromorphic function is a complex function that is holomorphic (analytic) on an open subset of the complex plane except for a set of isolated points, which are poles where the function can take infinite values. This means that meromorphic functions are allowed to have poles, but they are otherwise well-behaved and can be expressed as the ratio of two holomorphic functions.
Parameterization: Parameterization refers to the process of expressing a curve, surface, or other geometric object using parameters, which are typically variables that describe the object's position or shape. This concept is crucial for simplifying the evaluation of integrals, particularly in complex analysis, as it allows for the mapping of complicated paths or domains into simpler forms that are easier to work with. Through parameterization, integrals can be transformed into manageable forms that facilitate the application of various mathematical techniques.
Pole: In complex analysis, a pole is a type of singularity of a function where the function approaches infinity as the variable approaches a certain point. Poles are significant because they help identify the behavior of complex functions near these critical points and play a crucial role in calculating residues, which are used to evaluate integrals over closed contours.
Principal Value: The principal value refers to a specific interpretation of a mathematical expression, particularly in contexts where that expression may be multivalued or improperly defined, such as integrals or logarithmic functions. This concept is vital in resolving ambiguities that arise in complex analysis, ensuring that calculations yield a single, well-defined result. By selecting a principal value, one can avoid complications due to branch cuts or improper behavior at singularities.
Rational Function: A rational function is a function that can be expressed as the ratio of two polynomial functions, where the denominator is not zero. These functions can be represented in the form $$R(x) = \frac{P(x)}{Q(x)}$$, where $$P(x)$$ and $$Q(x)$$ are polynomials. Rational functions play a critical role in evaluating integrals using residues, especially when dealing with complex variables and contour integration.
Residue Theorem: The residue theorem is a powerful result in complex analysis that relates contour integrals of holomorphic functions around singularities to the residues at those singularities. It states that the integral of a function over a closed contour can be calculated by summing the residues of the function's singular points enclosed by the contour, multiplied by $2\pi i$. This theorem serves as a cornerstone for evaluating integrals and series in complex analysis and has broad applications in real integrals, physics, and engineering.
Residue Theory: Residue theory is a powerful tool in complex analysis that deals with the evaluation of integrals and the summation of series by analyzing singularities of analytic functions. It leverages the concept of residues, which are coefficients in the Laurent series expansion of a function around its singular points. This approach simplifies complex calculations by transforming difficult contour integrals into manageable sums involving residues at poles.
Singularity: In complex analysis, a singularity refers to a point at which a complex function ceases to be well-defined or analytic. Singularities are important because they help classify functions and determine their behavior, especially when dealing with integrals and residues in complex planes.
Substitution in Complex Variables: Substitution in complex variables refers to the method of transforming a complex integral into a simpler form by changing the variable of integration. This technique often involves mapping a complex function into a different domain, allowing for easier evaluation of integrals, particularly when dealing with singularities and residues. By using substitutions, complex analysis can simplify problems that might be difficult to tackle directly.
Trigonometric Integrals: Trigonometric integrals refer to integrals that involve trigonometric functions such as sine, cosine, tangent, and their respective powers. These integrals often require special techniques for evaluation, including substitution and the use of identities. Understanding how to handle trigonometric integrals is essential when applying the residue theorem to evaluate real integrals involving these functions.
Uniform Convergence: Uniform convergence is a type of convergence for sequences of functions where the speed of convergence is uniform across the entire domain. This means that for any chosen level of accuracy, there is a single point in the domain from which all functions converge uniformly, ensuring the limit function preserves continuity and other properties of the original functions.
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