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Replusz
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- TL;DR Summary
- I don't quite understand the steps taken to get to the second line (3.71->3.72):
(http://www.damtp.cam.ac.uk/user/tong/qft/qft.pdf)
Invert an integral equation to find this unknown
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- Thread starter Replusz
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In summary, to invert an integral equation and find an unknown, you need to isolate the unknown variable and perform the inverse operation of the integral. The purpose of inverting an integral equation is to solve for an unknown variable and better understand the relationship between the original function and the integral. There are limitations to inverting an integral equation, such as well-behaved functions and convergent integrals. An example of inverting an integral equation is solving for x in ∫(x+2) dx = 10. Some tips for successfully inverting an integral equation include carefully manipulating the equation, using algebraic techniques, and checking the solution for validity.
- #2
S.G. Janssens
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Apart from the insane habit in physics to write the measure in front of the integrand (
) and other conventions related to factors of ##\pi##, this follows from taking inverse Fourier transforms on both sides of (3.71).
) and other conventions related to factors of ##\pi##, this follows from taking inverse Fourier transforms on both sides of (3.71).- #3
Replusz
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Thank you for your quick and perfectly clear reply! :)
1. How do I invert an integral equation to find an unknown?
Inverting an integral equation involves using mathematical techniques such as integration by parts, substitution, or series expansion to manipulate the equation and solve for the unknown variable. It requires a strong understanding of calculus and algebraic manipulation.
2. What are the common methods used to invert an integral equation?
The most commonly used methods to invert an integral equation include Laplace transforms, Fourier transforms, and Mellin transforms. These methods involve transforming the integral equation into a simpler form that can be solved using algebraic techniques.
3. Can all integral equations be inverted to find an unknown?
No, not all integral equations can be inverted to find an unknown. Some equations may be too complex to be manipulated using traditional methods, while others may not have a unique solution. It is important to carefully analyze the equation and determine if it is possible to invert it.
4. Are there any limitations to inverting an integral equation?
Yes, there are limitations to inverting an integral equation. Some equations may have multiple solutions, making it difficult to determine the correct one. In addition, the process of inverting an integral equation can be time-consuming and may require advanced mathematical skills.
5. How can I check if my solution to an inverted integral equation is correct?
To check the correctness of your solution, you can substitute the value of the unknown variable back into the original integral equation and see if it satisfies the equation. You can also use numerical methods to approximate the solution and compare it to your calculated value.
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