Analytical Integration of a Difficult Function

In summary, the conversation discusses the possibility of integrating a given function analytically. The function represents a chemical reaction rate and contains real numbers A, B, and β. The question is whether the integral is finite and if there is a coordinate transformation that can make it bounded. It is suggested that the integral behaves well at infinity, but may have issues at zero. Further analysis is needed to determine the behavior near zero and at the tail from β to infinity.
  • #1
junt
18
1
Is it possible to integrate the following function analytically?

##\int_{0}^{\infty} \frac{\exp{-(\frac{A}{\tau}+B\tau+\frac{A}{\beta-\tau})}}{\sqrt{\tau(\beta-\tau)}}d\tau,##

where ##A##, ##B## and ##\beta## are real numbers. What sort of coordinate transformation makes the integral bounded? Is it even bounded? Are these poles integrable?

Any help is much appreciated!
 
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  • #3
jedishrfu said:
Can you provide a context of where you got this problem?
Integrals like this appear when one is looking at chemical reaction rates. The exponent is basically the classical action. A and B contains space coordinates, which will be integrated after integral over ##\tau## has been performed.
 
  • #4
Hi junt:

I think I understand that you are asking about whether the integral is finite. I think it is easy to see that the integrand behaves OK at infinity. It is a bit trickier to consider behavior at zero. Can you simplify the integrand behavior near zero and see if the integral of the simplification is OK? That is , consider the integral from zero to ε<<1 of a simplified integrand between zero and ε.

Regards,
Buzz
 
  • #5
I've played a bit with WolframAlpha and it suggests that between 0 and ##\beta## it should be fine but the tail from ##\beta## to ##\infty## is purely imaginary and also infinite.
 
  • #6
SlowThinker said:
is purely imaginary
Sure, since the expression under the root becomes negative
 

FAQ: Analytical Integration of a Difficult Function

What is analytical integration?

Analytical integration is a mathematical technique used to find the exact solution to an indefinite integral, or the reverse of differentiation. It involves finding an antiderivative of a function in order to solve for the original function.

How do you integrate a difficult function analytically?

To integrate a difficult function analytically, you must first identify any patterns or known integration techniques that can be applied. Then, you can use substitution, integration by parts, or other techniques to simplify the function and find its antiderivative. It may also be helpful to use software or tables to look up known integrals if necessary.

Why is analytical integration important?

Analytical integration is important because it allows us to find exact solutions to integrals, which are used in many areas of mathematics and science. It also allows us to understand the behavior and properties of functions, and can be applied in the development of new theories and models.

What are some common challenges in analytical integration?

Some common challenges in analytical integration include dealing with complex or highly variable functions, identifying appropriate integration techniques, and handling functions with multiple variables. It can also be difficult to find closed-form solutions for some integrals, requiring the use of numerical methods instead.

Can analytical integration be used in real-world applications?

Yes, analytical integration is widely used in real-world applications, especially in fields such as physics, engineering, economics, and statistics. It is used to solve problems involving rates of change, optimization, and probability, among others. It is also used in the development of mathematical models and simulations.

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