Lebesgue/Cauchy-Riemann Integral

  • Thread starter lozzlepop
  • Start date
  • Tags
    Integral
In summary, the given problem involves evaluating the Cauchy-Riemann integral of \intx^{2k+1}ln(x) dx from 0\leqx\leq1 for a non-negative integer K. To solve this, it is necessary to prove that the modulus of the function is Lebesgue integrable and then use the Monotone Convergence Theorem and integration by parts to find the integral. Further assistance is needed in proving the Lebesgue integrability of the modulus.
  • #1
lozzlepop
1
0

Homework Statement



Let K be a non-negative integer. Evaluate the Cauchy-Riemann integral
[tex]\int[/tex]x[tex]^{2k+1}[/tex]ln(x) dx from 0[tex]\leq[/tex]x[tex]\leq[/tex]1

Homework Equations





The Attempt at a Solution


so far I've got as far as working out that I need to check that this is Lebesgue-integrable, then the Lebesgue integral over the domain is equal to the Cauchy-Riemann integral. However the function is negative over the domain. I think that this means that if I find the modulus of the function and can prove that it is Lebesgue integrable then the function is Lebesgue integrable and then can use the Monotone Convergence Theorem and integration by parts to find the integral.
 
Physics news on Phys.org
  • #2
However I'm not sure how to prove that the modulus is Lebesgue integrable. Any help would be appreciated.
 

1. What is the difference between Lebesgue and Cauchy-Riemann Integral?

The Lebesgue Integral and Cauchy-Riemann Integral are two different methods of calculating the integral of a function. The Lebesgue Integral is a more general form of integration that allows for a wider range of functions to be integrated, while the Cauchy-Riemann Integral is a specific form of integration used for complex-valued functions.

2. How does the Lebesgue Integral handle non-continuous functions?

Unlike the Riemann Integral, the Lebesgue Integral can handle non-continuous functions. This is because the Lebesgue Integral uses a different approach in defining the integral, by considering the area under the curve as the limit of a sum of smaller areas instead of the limit of a sum of infinitely small rectangles.

3. What is the significance of the Cauchy-Riemann equations in the Cauchy-Riemann Integral?

The Cauchy-Riemann equations are a set of necessary and sufficient conditions for a complex-valued function to be differentiable at a point. These equations are crucial in the Cauchy-Riemann Integral as they allow for the calculation of the integral by using only the values of the function at the boundary points of the region of integration.

4. Can the Lebesgue Integral be applied to infinite intervals?

Yes, the Lebesgue Integral can be applied to infinite intervals as long as the function is Lebesgue integrable. This means that the function must be measurable and its integral over the entire domain is finite. The Lebesgue Integral also allows for the integration of unbounded functions.

5. What are the advantages of using the Cauchy-Riemann Integral over the Lebesgue Integral?

The Cauchy-Riemann Integral has some advantages over the Lebesgue Integral in certain cases. One of the main advantages is that it allows for the evaluation of complex-valued integrals without having to consider the real and imaginary parts separately. Additionally, the Cauchy-Riemann Integral can be used to calculate integrals over regions with holes, which is not possible with the Lebesgue Integral.

Similar threads

A Compute Lebesgue integral as (improper) Riemann integral
    • Topology and Analysis
Replies
5
Views
2K
Contour Integration over Square, Complex Anaylsis
    • Calculus and Beyond Homework Help
Replies
3
Views
2K
Prove that an Lebesgue integral exists iff the other exists
    • Calculus and Beyond Homework Help
Replies
6
Views
1K
Where is ##(z+1)Ln(z)## differentiable?
    • Calculus and Beyond Homework Help
Replies
5
Views
997
Understanding the Cauchy Integration Formula for Analytic Functions
    • Calculus and Beyond Homework Help
Replies
4
Views
1K
Prove limit comparison test for Integrals
    • Calculus and Beyond Homework Help
Replies
2
Views
842
Measurable with Respect to Complete Space
    • Calculus and Beyond Homework Help
Replies
1
Views
1K
Proving Equality Using Lebesgue Monotone Convergence Theorem
    • Calculus and Beyond Homework Help
Replies
2
Views
1K
Need a hint proving that integral converges
    • Calculus and Beyond Homework Help
Replies
2
Views
1K
Determining Cauchy principal value of divergent integrals
    • Calculus and Beyond Homework Help
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top