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Related Concept Videos

Improper Integrals: Discontinuous Integrands01:28

Improper Integrals: Discontinuous Integrands

Evaluating Areas Under Curves with DiscontinuitiesA definite integral is considered improper when the integrand is discontinuous at one of the limits of integration. This occurs when the function is undefined or becomes infinite at an endpoint, making the corresponding region under the curve unbounded. Such behavior is commonly associated with vertical asymptotes at the boundary of the interval. To properly define and evaluate these integrals, a limiting process is used to determine whether a...
Improper Integrals: Infinite Intervals01:29

Improper Integrals: Infinite Intervals

An integral is classified as improper due to an infinite interval when at least one of its limits of integration extends to positive or negative infinity. In such cases, the region under the curve is unbounded, and standard techniques for evaluating definite integrals are not directly applicable. Instead, the improper integral is defined through a limiting process that allows one to determine whether the accumulated area remains finite despite the infinite domain.Application to Exponential...
Integration by Parts: Indefinite Integrals01:26

Integration by Parts: Indefinite Integrals

Integration by parts is a fundamental technique in calculus for evaluating integrals involving the product of two functions. It is particularly useful when direct integration is not feasible. The method is based on the product rule for differentiation, which states that the derivative of a product equals the derivative of the first function times the second, plus the first function times the derivative of the second. By integrating this identity and rearranging terms, the integration by parts...
Integration by Parts: Definite Integrals01:23

Integration by Parts: Definite Integrals

Definite integrals involving the product of two functions over a fixed interval can be evaluated using integration by parts. This method rewrites the integral as the difference of a product evaluated at the endpoints and a remaining definite integral that is often simpler to compute.A representative example is the definite integral of the inverse tangent function. Since there is no direct integration formula for arctan ⁡x, the integrand is rewritten as a product of arctan⁡ x and the constant...
Indefinite Integrals01:25

Indefinite Integrals

The water inflow rate into a storage tank is not constant but increases over time. Initially, the pump delivers water at a rate of 5 L/min. However, the inflow rate increases by 2 L/min for each additional minute due to rising pressure or system adjustments. This scenario can be described mathematically by a linear function:It is necessary to integrate the inflow rate function to measure the total volume of water added to the tank over time. The total water volume V(t) is obtained by performing...
Substitution Rule Applied to Indefinite Integrals01:27

Substitution Rule Applied to Indefinite Integrals

When a force is applied to a linear spring, the restoring force increases proportionally with the amount of displacement. This behavior is described by Hooke’s law, which allows the work done on the spring to be determined directly from the force–displacement relationship. In this case, the force varies in a simple and predictable manner, making the calculation relatively simple.On the other hand, a nonlinear spring does not obey Hooke’s law. Its restoring force depends on position in a...

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Setting Limits on Supersymmetry Using Simplified Models
07:46

Setting Limits on Supersymmetry Using Simplified Models

Published on: November 15, 2013

Taming hypersingular integrals using dimensional continuation.

Zehao Li1, L R Ram-Mohan

  • 1Department of Physics, Worcester Polytechnic Institute, Worcester, Massachusetts 01609, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|March 10, 2012
PubMed
Summary
This summary is machine-generated.

This study introduces dimensional continuation to handle hypersingular integrals in boundary integral techniques. This method simplifies computations in potential theory, electromagnetics, and mechanics by isolating and ignoring singularities.

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Area of Science:

  • Computational Mathematics
  • Applied Physics
  • Continuum Mechanics

Background:

  • Boundary integral techniques often involve hypersingular integrals that complicate computations.
  • Existing methods may require complex analysis or contour distortions to manage these singularities.

Purpose of the Study:

  • To develop a method for isolating and managing hypersingular integrals in boundary integral equations.
  • To simplify the evaluation of integrals in potential theory, electromagnetic scattering, and crack dynamics.

Main Methods:

  • Dimensional continuation is employed to isolate singularities in integrals of Green's functions.
  • New forms of the Dirac delta function in D dimensions are identified.
  • A theorem is presented to reduce 3D hypersingular integrals to 1D finite integrals.

Main Results:

  • Rules for extracting the finite part of hypersingular integrals are established.
  • Computational techniques are simplified, avoiding complex analysis and contour distortions.
  • Hypersingular integrals in 3D potential problems are reduced to one-dimensional finite integrals.

Conclusions:

  • The dimensional continuation method provides a consistent and efficient way to evaluate boundary integrals.
  • This approach significantly enhances the applicability of computational techniques in science and engineering.
  • The method offers a direct evaluation of integrals, validated by comparison with existing results.