Trying to find integral using laplace method

In summary, the conversation discusses integrating the function exp(N(lns-s)) with respect to s for different values of x using the Laplace method. The maximum point is at x=1 and the answer is known for x<1 and x=1, but there is struggle for x>1 as the stationary point is no longer inside the interval. The conversation also mentions transforming the equation into a Gamma function and using the incomplete gamma function to solve for x>1.
  • #1
Mathsone
3
0
∫∞x exp(N(lns-s)) ds

how do integrate this when x>1 x<1 and x=1 using laplace method?
the maximum point is at x=1


i have the answer for x<1 and x=1

but I am struggling for x>1 as the stationary point is no longer inside the interval.
 
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  • #2
Hey Mathsone and welcome to the forums.

In your integral x is not being integrated with respect to which means it is a constant which doesn't have any effect on the integral.

Is the 'x' meant to be an 's' or the 's' meant to be an 'x'? (Both end up being the same thing).
 
  • #3
the integral is from x to infinity, the function is exp(N(lns-s)) which is being integrated with respect to s.

i think the format of my integral just got meddled up when i was typing out the question. sorry bout that.
 
  • #4
Do you have a particular value for N or do you have to show that it can or can't be solved for some appropriate N (like integers).

For a start your expression simplifies down to s^N x e^(-sN) which is nearly in the form of a Gamma function. I have transformed your equation into a gamma function with a simple transformation using the substituion t = nS and then changing the limits to get it into a Gamma function.

Using this information, does this help you with your question?
 
  • #5
N is a large integer. hm...well yes, but i did do that. its jus that i don't understand how to calculate the integral when x>1
 
  • #6
Mathsone said:
N is a large integer. hm...well yes, but i did do that. its jus that i don't understand how to calculate the integral when x>1

Are you aware of the incomplete gamma function?
 

FAQ: Trying to find integral using laplace method

What is the Laplace method for finding integrals?

The Laplace method is a mathematical technique used to evaluate integrals that involve complex functions or infinite intervals. It involves transforming the integral into a complex plane and using the residue theorem to solve it.

When is the Laplace method useful?

The Laplace method is useful when dealing with integrals that cannot be solved using traditional methods, such as those involving complex functions or infinite intervals. It is also helpful in finding asymptotic approximations of integrals.

How do you use the Laplace method to find integrals?

To use the Laplace method, you first need to transform the integral into a complex plane by substituting the integral variable with a complex variable. Then, you can use the residue theorem to evaluate the integral. Finally, you can convert the solution back to the original variable to get the final result.

What are the limitations of the Laplace method?

The Laplace method is not always applicable, as it requires the function to be analytic and have a finite number of singularities. Additionally, the integral must have a path of integration that can be deformed into a closed contour in the complex plane.

Are there any real-world applications of the Laplace method?

Yes, the Laplace method has various real-world applications, including in physics, engineering, and economics. It can be used to solve problems involving oscillatory integrals, probability distributions, and differential equations. It is also used in signal processing and control systems.

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