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?
 

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

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.

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.

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.

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.

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

I On convolution theorem of Laplace transform: Schiff
Replies
4
Views
2K
I Gaussian integral by differentiating under the integral sign
Replies
19
Views
3K
A How do I apply the method of steepest descents to this integral?
Replies
1
Views
969
I Solving the Difficult Integral ##\int_0^{\infty} x^{n+1} e^{-x} \sin(ax) dx##
Replies
5
Views
2K
A Multiplying divergent integrals using Hardy fields approach
Replies
3
Views
2K
A A question about a complex integral
Replies
8
Views
2K
A Summing over continuum and uncountable numerocities
Replies
1
Views
2K
I Integrating a product of exponential and trigonometric functions
Replies
21
Views
2K
Complex integral in finite contour at semiaxis
Replies
1
Views
1K
I Why wasn't this symbol "swapped"?
Replies
2
Views
1K

Hot Threads

Recent Insights

Back
Top