Trying to find integral using laplace method

[Mathsone]

∫∞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.

 

[chiro]

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).

 

[Mathsone]

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.

 

[chiro]

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?

 

[Mathsone]

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

 

[chiro]

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?