How to Integrate e^x / (1+x^2) with Complex Contour Integration?

In summary, the integral of e^x / (1+x^2) is a complex integral that cannot be expressed in terms of elementary functions and requires advanced techniques to solve. It can be solved using numerical methods on a computer or calculator, but the exact solution cannot be obtained. This integral is significant in mathematics as it is used in various applications and is also a fundamental integral in the study of complex analysis. It has real-world applications in calculating areas under curves, studying damping in oscillatory systems, and calculating electric fields in physics.
  • #1
Einsling
2
0
Hi
Could you please tell me how to integrate it? Thanks ~~

[tex]\int_{-\infty}^{\infty} \frac {e^{i\omega t}} {1+\frac{t^2}{\tau^2}} dt[/tex]

where i is imaginary unit, [tex]\omega, \tau[/tex] are positive real,
 
Last edited:
Physics news on Phys.org
  • #2
Complex contour integration should be good. Are you familiar with the residue theorem from complex variables?
 

1. What is the integral of e^x / (1+x^2)?

The integral of e^x / (1+x^2) is a complex integral that cannot be expressed in terms of elementary functions. It can be solved using techniques such as substitution or integration by parts.

2. Is there a shortcut or formula for solving this integral?

No, there is no simple formula for solving this integral. It requires advanced techniques and manipulation to obtain a solution.

3. Can this integral be solved using a computer or calculator?

Yes, this integral can be solved using numerical methods on a computer or calculator. However, the exact solution cannot be obtained as it involves infinite series.

4. What is the significance of this integral in mathematics?

The integral of e^x / (1+x^2) is an important integral in calculus and is used in various applications such as probability, physics, and engineering. It is also a fundamental integral in the study of complex analysis.

5. Are there any real-world applications of this integral?

Yes, this integral has many real-world applications, such as in the calculation of areas under curves, in the study of damping in oscillatory systems, and in the calculation of electric fields in physics.

Similar threads

Replies
4
Views
609
Replies
1
Views
793
Replies
3
Views
1K
Replies
19
Views
2K
Replies
21
Views
639
  • Calculus
Replies
8
Views
2K
Replies
2
Views
795
Replies
2
Views
1K
Replies
2
Views
911
Back
Top