Improper integrals with parameters

In summary, we are trying to prove that the limit of F(u) as u approaches 0 is equal to f(0), given that f is continuous on the interval [0,inf] with absolute value less than M. To do this, we split the integral into the intervals (0,1) and (1,inf), and use the fact that the interval (1,inf) is controlled. We can then split the integral further into (0,e) and (e,inf) and show that for any e>0, we can choose u close enough to zero so that the second term goes to zero. Finally, we consider the limit as e approaches 0 using the continuity of f.
  • #1
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Homework Statement



Let f be continuous on the interval [0,inf] with abs(f(x)) < M, set
F(u) = [tex]\int[/tex] [tex]\frac{u*f(x)}{u^2 + x^2}[/tex] integral from 0 to inf (bad with latex)

prove that lim F(u) as u --> 0 = f(0)

Homework Equations


none

The Attempt at a Solution



I split the integral into the intervals from (0,1) and (1,inf) since (1,inf) is controlled

now I can't seem to work out the first term. Any help would be great as I'm totally stuck
 
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  • #2
qualitatively, how about splitting it into (0,e) and (e,inf), then show for any e>0, you can choose u close enough to zero that the 2nd term goes to zero, then consider the limit as e->0 using the continuity of f
 

FAQ: Improper integrals with parameters

What is an improper integral with parameters?

An improper integral with parameters is an integral that contains a variable, or parameter, in the limits of integration. This means that the limits of integration are not fixed numbers, but instead can change depending on the value of the parameter.

Why do we use parameters in improper integrals?

Parameters are used in improper integrals to allow for more flexibility in the calculation. By including a parameter, we can integrate a wider range of functions and also account for different scenarios or situations.

What are the challenges of working with improper integrals with parameters?

One of the main challenges of working with improper integrals with parameters is that the limits of integration may not always be well-defined. This can lead to difficulties in evaluating the integral and may require additional techniques, such as limits or substitution, to solve. Additionally, the presence of parameters can make the integral more complex and may require advanced mathematical skills to solve.

How do you solve an improper integral with parameters?

The general approach to solving an improper integral with parameters involves first identifying the type of improper integral (infinite limits, discontinuous integrand, etc.), then manipulating the integral to separate out the parameter, and finally evaluating the integral using appropriate techniques, such as substitution or partial fractions.

What are some real-world applications of improper integrals with parameters?

Improper integrals with parameters have many real-world applications in fields such as physics, economics, and engineering. For example, in physics, parameters can represent physical constants or variables in equations, and improper integrals can be used to calculate important quantities such as center of mass or moments of inertia. In economics, parameters can represent variables in economic models, and improper integrals can be used to calculate quantities such as total cost or revenue.

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