## Improper Integrals I

We’ve dealt with Riemann integrals and their extensions to Riemann-Stieltjes integrals. But these are both defined to integrate a function over a finite interval. What if we want to integrate over an infinite ray, like all positive numbers?

As a specific example, let’s consider the function , and let it be defined on the ray . For any real number we can pick some . In the interval the function is continuous and of bounded variation (in fact it’s decreasing), and so it’s integrable with respect to . Then it’s integrable over the subinterval . Why not just start by saying it’s integrable over ? Because now we have a function on defined by

Since is differentiable and is continuous at , we see that is differentiable here, and its derivative is . This result is independent of the we picked.

Since we can do this for any we get a function defined for . Its derivative must be , and we can check that also has this derivative, so these two functions can only differ by a constant. Clearly we want , since at that point we’re “integrating” over a degenerate interval consisting of a single point. This fixes our function as .

Now our question is, what happens as we take to get larger and larger? Our intervals get bigger and bigger, trying to fill out the whole ray . And for each one we have a value for the integral: . So we take the limit as approaches infinity: . This will be the value of the integral over the entire ray.

We turn this rubric into a definition: given a function that is integrable with respect to over the interval for all , we can define a function on by

We define the improper integral to be the limit

if this limit exists. Otherwise we say that the integral diverges.

We can similarly define improper integrals for leftward rays as

And over the entire real line by choosing an arbitrary point and defining

That is, we take the two bounds of integration to go to their respective infinities *separately*. It must be noted that the limit where they go to infinity together:

may exist even if the improper integral diverges. In this case we call it the “Cauchy principal value” of the integral, but it is *not* the only justifiable value we could assign to the integral. For example, it’s easy to check that

so the Cauchy principal value is . However, we might also consider

which diverges.

[…] Tests for Improper Integrals We have a few tests that will come in handy for determining if an improper integral converges. In all of these we’ll have an integrator on the ray , and a function which is […]

Pingback by Convergence Tests for Improper Integrals « The Unapologetic Mathematician | April 21, 2008 |

[…] on the ray , and be any function integrable with respect to through the whole ray. Then if the improper integral converges, then so does […]

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Is there some significance attached with the Cauchy Principal Value? I mean since it has been given a name, it most have encountered at more places.

Comment by Nilay | April 23, 2008 |

I’m not sure what I’ll do with it yet, but I may as well give its name now, and if it comes up later I can refer back to it.

Comment by John Armstrong | April 23, 2008 |

[…] the particular case of an improper integral, we have . Then . Our condition then […]

Pingback by Cauchy’s Condition « The Unapologetic Mathematician | April 23, 2008 |

[…] Then as we let go to infinity, goes to infinity with it. Thus the sum of the series is the same as the improper integral. […]

Pingback by Infinite Series « The Unapologetic Mathematician | April 24, 2008 |

[…] can forge a direct connection between the sum of an infinite series and the improper integral of a function using the famed integral test for […]

Pingback by The Integral Test « The Unapologetic Mathematician | April 29, 2008 |

[…] I get the title of the post wrong? No. It turns out that I covered half of this topic two years ago as I prepared to push into infinite series. Back then, I dealt with what happened when we wanted to […]

Pingback by Improper Integrals II « The Unapologetic Mathematician | January 15, 2010 |