Integrating odd functions with infinite discontinuity:

In summary, the conversation discusses the integration of odd functions with infinite discontinuities in their domain. It is suggested that these types of integrals can be simplified and the infinite discontinuities removed. The concept of improper integrals is mentioned and the question of whether these integrals will converge or diverge is raised. The idea of Cauchy Principal Value is brought up as a way to assign a value to these divergent integrals. The conversation concludes with appreciation for the insight provided.
  • #1
atqamar
55
0
If an odd function has an infinite discontinuity in its domain, can it be integrated (such that a convergent finite emerges) with that domain included?

For example: [tex]\int_{-1}^2 \frac{1}{x^{-3}} dx[/tex]. Intuitively, it can be simplified to [tex]\int_1^2 \frac{1}{x^{-3}} dx[/tex] and thus the infinite discontinuity at 0 is removed.

If that is not doable, can an integral converge if the end points of the domain are infinite discontinuities?

For example: Does [tex]\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} tan(x) dx = 0[/tex]?

If these kinds of functions are split in two, and limits to [tex]\infty[/tex] are taken, then algebraic manipulation of infinities are required.

Any insight would be appreciated.
 
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  • #2
You should look into improper integrals?
 
  • #3
Certainly, these are all improper integrals. But my question is whether the above integrals would converge or diverge. Intuitively/Geometrically, they should converge; but like I said, once the improper integral is carried out, one has to deal with infinities.
 
  • #4
Look up "Cauchy Principal Value". Strictly, the integrations over the discontinuities are divergent, but they can be assigned a Cauchy Principal Value that appeals to your geometric intuition.
 
  • #5
Thanks a lot Gib Z! That helped immensely.
 

Related to Integrating odd functions with infinite discontinuity:

What is an odd function?

An odd function is a mathematical function that satisfies the property f(-x) = -f(x) for all values of x. This means that when the input to the function is negated, the output is also negated. Graphically, an odd function will have symmetry about the origin.

What is an infinite discontinuity?

An infinite discontinuity occurs when a function approaches either positive or negative infinity at a certain point. This can happen when the function has a vertical asymptote or when the limit of the function at a certain point does not exist.

Why is it important to integrate odd functions with infinite discontinuity?

Integrating odd functions with infinite discontinuity is important because it allows us to find the area under the curve of these types of functions. This can be useful in many applications, such as calculating work or finding the center of mass of an object.

What techniques can be used to integrate odd functions with infinite discontinuity?

One technique that can be used is called the Cauchy principal value, which involves taking the limit of the integral as the limits of integration approach the point of the infinite discontinuity. Another technique is to split the integral into two parts, one on each side of the point of discontinuity, and then take the limit of the integral as the limits of integration approach the point.

Can odd functions with infinite discontinuity be integrated using traditional integration methods?

In general, traditional integration methods such as the power rule or substitution may not work for odd functions with infinite discontinuity. However, these functions can still be integrated using the techniques mentioned above, or by using numerical methods such as Simpson's rule or the trapezoidal rule.

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