- #1
Integrating Hard Integrals: No Closed Form Solution
- Thread starter semigroups
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In summary, the conversation is about a given integral that converges but does not have a closed form solution. The speaker suggests changing the region of integration to simplify the problem, but the person being addressed has already tried this and is unable to find a solution. The speaker then asks for more details on what the person tried.
- #2
chiro
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semigroups said:
Hey semigroups and welcome to the forums.
Have you considered changing the region of integration to get things in terms of one variable?
My guess is if this is a problem from a problem set, that a change of variables and some extra stuff will give you something that ends up being in the form of a Beta (complete or incomplete) or a Gamma (complete or incomplete) and then you can just leave it at that.
If you have tried changing the region (I think it should look something like a triangle in the [0,1]x[0,1] region) then what did you try exactly and what did you find out?
- #3
1. What is meant by a "hard integral"?
A hard integral is a mathematical expression that cannot be solved using traditional methods, such as substitution or integration by parts. These integrals often involve complex functions or do not have a standard form that can be easily integrated.
2. Why are hard integrals difficult to solve?
Hard integrals are difficult to solve because they do not have a closed form solution, meaning there is no simple expression that represents the integral. This makes it challenging to find an exact solution and requires more advanced techniques to approximate the integral.
3. What are some techniques for integrating hard integrals?
Some techniques for integrating hard integrals include using numerical methods, such as the trapezoidal rule or Simpson's rule, to approximate the integral. Other techniques include using series expansions, contour integration, or special functions such as the Gamma function.
4. Are there any real-world applications for integrating hard integrals?
Yes, there are many real-world applications for integrating hard integrals. For example, in physics, hard integrals are often used to calculate the trajectory of a projectile or the area under a curve in a force vs. time graph. In economics, hard integrals can be used to model complex economic systems. They are also used in engineering, statistics, and many other fields.
5. Is there ongoing research in the field of hard integrals?
Yes, there is ongoing research in the field of hard integrals as they are a fundamental part of mathematics and have many practical applications. Researchers are constantly developing new techniques and algorithms to solve hard integrals more efficiently and accurately.
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