Can we give a meaning to this integral

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  • #1
mhill
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Let be the integral

[tex] g(x)= \int_{-\infty}^{\infty}dy \frac{f(y)}{ |x-y|^{1/2}} [/tex]

for given values of 'x' does it mean anything ? , let's take into account that no matter what value of 'x' we chose there is always a singularity at y=x so the expression above would be always divergent , here |x| means absolute value of x or the modulus (in case x is a complex number)
 
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  • #2
You can understand it as a Cauchy Principal Value integral, which is possibly infinite.
 

1. What is the purpose of finding the meaning of an integral?

The purpose of finding the meaning of an integral is to understand the relationship between the function being integrated and the area under its curve. This can help in solving various real-world problems and making predictions in different fields such as physics, engineering, and economics.

2. Can we always give a meaning to every integral?

Not necessarily. Some integrals may not have a closed form or may be too complex to find a simple meaning. In these cases, we can still approximate the integral using numerical methods, but the exact meaning may not be possible to determine.

3. How do we give a meaning to an integral?

The most common way to give a meaning to an integral is by interpreting it as the area under the curve of the function being integrated. This can be visualized using a graph or geometrically using shapes such as rectangles or trapezoids. Other possible interpretations include the accumulation of a quantity over time or the average value of a function.

4. Are there any other applications of integrals besides finding area?

Yes, integrals have various applications in different fields. They can be used to calculate volume, displacement, work, and even probability. In physics, integrals are used to find the center of mass, moment of inertia, and other important physical quantities. In economics, integrals are used to calculate consumer and producer surplus.

5. Can we give a meaning to improper integrals?

Yes, improper integrals can also have a meaningful interpretation in certain cases. For example, an improper integral with a lower limit of zero can represent the area under a curve that extends to infinity. Other improper integrals may represent the average value of a function over an infinite interval. However, some improper integrals may not have a meaningful interpretation and can diverge to infinity.

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