Improper Integral of the following integral?

Click For Summary

Discussion Overview

The discussion revolves around the computation of an improper integral using residues, specifically the integral of exp(x + 1/x)/x over the limits [0, ∞). Participants explore the convergence of the integral and the application of complex analysis techniques.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant asks how to compute the improper integral using residues.
  • Another participant asserts that the integral does not converge, suggesting that there is no point in attempting to prove this using residues.
  • A different participant expresses confusion about the convergence, noting that their software indicates a convergence to approximately 0.2278, despite the previous claim.
  • This participant describes their approach using residues, integrating along a contour and applying the binomial theorem to analyze the series representation of the function.
  • Another participant argues that for positive x, the integrand exp(x + 1/x) is bounded below by e^x/x, which diverges as x approaches infinity, implying that the integral cannot have finite area.
  • A later post corrects the integrand to exp[-(x + 1/x)]/x, indicating a potential misunderstanding in the original formulation.

Areas of Agreement / Disagreement

Participants express disagreement regarding the convergence of the integral, with some asserting it diverges and others suggesting it converges based on computational results. The discussion remains unresolved.

Contextual Notes

Limitations include the initial misstatement of the integrand, which may affect the analysis of convergence and the application of residues. The discussion also reflects differing interpretations of the behavior of the integral at infinity.

eyesontheball1
Messages
31
Reaction score
0
How would one go about computing the following improper integral, with limits of integration [0,∞) using residues?

[itex]\int exp(x+1/x)/x[/itex]
 
Physics news on Phys.org
The integral doesn't converge. There's no point in trying to find a way to prove that using residues.
 
How does it not converge though? I've been confused by this integral for quite some time. Using some crappy software of mine, the integral seems to converge to .2278. However, when I apply residues to this improper integral and integrate along a contour with a counter-clockwise semi-circle in the upper half plane and a small circle with radius approaching 0 around the essential singularity at z=0, I get that the integral is equal to 0, by using the fact that the series representation of this function can be written as the product of two absolutely convergent sums, thus allowing one to use the binomial theorem to write out each summand and then rearrange the terms of the series to arrive at the result of the coefficient of 1/z being equal to the infinite series from n=0 to infinity of a_k, where a_k = kth term of the series = (x^k/k!)^2. Sorry about being messy, I'm kind of in a rush at the moment and don't have time to write out all the appropriate syntax.

David
 
eyesontheball1 said:
How does it not converge though?
For positive ##x##, ##\exp(x + 1/x) \geq \exp(x)##, so the integrand is bounded below by ##e^x/x##. The latter function diverges to infinity as ##x \rightarrow \infty##, so certainly it can't have finite area.
 
Oh WOW! I'm sorry guys. The integrand should be exp[-(x+1/x)]/x, heh.
 
Last edited:

Similar threads

MHB Improper integral of an even function
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Evaluation of an improper integral leading to a delta function
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Is interchanging the order of the surface and volume integrals valid here?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Improper Integrals: Definite & Indefinite | Bounds -1 to 1
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad How Can We Simplify Evaluating the Improper Integral of Logarithmic Functions?
  • · Replies 10 ·
Replies
10
Views
2K
Graduate Multiplying divergent integrals using Hardy fields approach
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad How to Approach a Double Exponential Integral?
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Integral Bee Preparation -- Trouble with this beautiful integral
  • · Replies 10 ·
Replies
10
Views
3K
Undergrad How does the limit comparison test for integrability go?
  • · Replies 2 ·
Replies
2
Views
2K
High School Integration by parts of inverse sine, a solved exercise, some doubts...
  • · Replies 4 ·
Replies
4
Views
3K