Finding the diversity order of Gamma function

Questions
1

I would like to simplify the following expression of Gamma function to obtain the diversity order (d) using Mathematica.

We assume that (\[Alpha]/s) \[LessLess] 1

d = Limit[-(Log[(2 - ((2 Gamma[ m, ((m*(\[Alpha]/s)^(1/n))/\[CapitalOmega])])/Gamma[m]))^ M])/Log[s], s -> \[Infinity]]

Thanks for your support!!

2

The expression “d” isn’t being evaluated. This means that the limit , as it is written now, must be hard to compute or doesn’t have an easy solution.

If you haven’t already tried it, please try using the Assumptions option for Limit. For example, you can try assuming that the parameters other than “s” are Real numbers or that they are positive. It is possible that these assumptions are needed to make the problem solvable:

Other than solving the limit, I am not sure what you are doing. If you can explain further, this might help us.

3

Thanks for your reply. However, the original function is (2 - ((2 Gamma[ m, ((m*(?/s)^(1/n))/?)])/Gamma))^ M , and I want first to simplify it by assuming (?/s) ? 1 , which means that (s) is very high positive value, thus how can we approximate this function when the condition (?/s) ? 1 is met?. By doing so, we can solve the limit.

4

There really isn’t a way to clearly convey (a/2)<<1 to the Limit function since it isn’t reall a rigorous statement. Are you trying to approximate the value of the limit? If this limit has a closed solution, I haven’t found it yet.

The best way to convey what you would like to do with this limit might be to come up with a simpler example of the same kind of problem that you know how to solve and see how you would solve it in Mathematica.

5

Could the Limit be preformed in pieces? (Mathematicians might correct this.)

Here I take the innermost section, take its limit as s-> Infinity, then stick in in
in place of the original counterpart.

In[2]:= Limit[-(2 Gamma[m, ((m*(\[Alpha]/s)^(1/n))/\[CapitalOmega])]),              
          s -> Infinity,                                                             
             Assumptions -> {n > 0, m > 0 , 0 < \[Alpha] < s,                           
            M > 0, \[CapitalOmega] > 0}]                                                

Out[2]= -2 Gamma[m]

In[3]:= d=Limit[-(Log[(2-((2 Gamma[m])/Gamma[m]))^M])/Log[s],  s->Infinity,            
           Assumptions->{n>0, m>0 , 0<\[Alpha] <s, \[CapitalOmega]>0}] 
Out[3]= Indeterminate

It appears that one ends up with

Log[0] / Log[Infinity]
6

Finding the diversity order of Gamma function