Hi guys:
I was reading a paper as I upload in the attachment. I tried to calculate Example 3.9 by hand, but ended with a different expression. So I started to doubt whether paper presents a right answer. Roughly, the question is based on the following:
The solution give in the paper is: (started with definition of 2F1)
I uploaded the notebook in the attachment too, if it could be convenient for you to have a look.
Really appreciate someone could help me with that. Do not hesitate to send me an email if that is more convenient: zhehaoz1@student.unimelb.edu.au
Just a first attempt at fixing the syntax problems:
$Assumptions = \[CapitalTheta] > 0 && t > 0 && \[Delta] > 0 &&
s < \[CapitalTheta];
Subscript[m, d][t_] =
Subscript[\[Lambda], 2] t + (
Subscript[\[Lambda], 1] - Subscript[\[Lambda], 2])/
Subscript[\[Lambda], 1] (1 - E^(-Subscript[\[Lambda], 1] t));
Subscript[M, X][s_] =
Integrate[
E^(s x) \[CapitalTheta]*E^(-\[CapitalTheta] x), {x, 0, Infinity}]
Subscript[M, Subscript[Z, 0][t_]][s_] = E^(
Subscript[\[Lambda], 2]
Integrate[
Subscript[M, X][s E^(-\[Delta] v)], {v, 0,
t}]) (*Equation (3.6)*)
Subscript[M, Subscript[Z, d][t_]][s_] =
1 + Integrate[(Subscript[M, X][s E^(-\[Delta] v)] - 1) Subscript[M,
Subscript[Z, 0] (t - v)][sE^(-\[Delta] v)] D[Subscript[m, d][v],
v], {v, 0, t}]
Dear Professor:
The result is still in the integral form. Is there any way to go further? Say, get the result in the Hyper-geometric function.
I suspect it may be confused by the
D[Subscript[m, d][v],v]
at the end of your integral.
While you might mathematically understand what the integral of ln(x) dsin(x^3) means, I think if you look at all the documentation and examples I think you will see they all show integration with respect to a simple variable, not a function and not a subscripted variable.
Can you express your integral in such a form and try evaluating the integration again?
Sorry, I was just trying to write a syntactically correct input. I still don’t know if it is what you wanted, and how to get a useful output.
Written this way we get Hypergeometric functions:
$Assumptions = \[CapitalTheta] > 0 && t > 0 && \[Delta] > 0 &&
s < \[CapitalTheta];
Subscript[m, d][t_] =
Subscript[\[Lambda],
2] t + (Subscript[\[Lambda], 1] - Subscript[\[Lambda], 2])/
Subscript[\[Lambda], 1] (1 - E^(-Subscript[\[Lambda], 1] t));
Subscript[M, X][s_] =
Integrate[
E^(s x) \[CapitalTheta]*E^(-\[CapitalTheta] x), {x, 0, Infinity}]
Subscript[M, Subscript[Z, 0][t_]][s_] =
E^(Subscript[\[Lambda], 2] Integrate[
Subscript[M, X][s E^(-\[Delta] v)], {v, 0, t}])
Integrate[(Subscript[M, X][s E^(-\[Delta] v)] - 1) Subscript[M,
Subscript[Z, 0] [t - v]][s E^(-\[Delta] v)] D[Subscript[m, d][v],
v], v]
Dear Professor:
Thanks for such elegant coding. i am wondering is there any way to track how Mathematica get the parameters of hpergeometric function, i.e show the calculation process. I tried “Trace”, however, it does not give me anything.
Sorry, I am no expert in special functions and integration.
