Solving for variable with Gamma and Exponential Function

Question

I am trying to solve for the dl parameter knowing all the remaining variables and as t->Infinity or very large number

m = 1.000001; a = 0.566837276858827; q1 = 35931.22171;
(E^(dl+a dl^(-1+m) Gamma[1-m,dl]-a dl^(-1+m) Gamma[1-m,dl t]) q1)/a ==25000

Mathematica responds with unsolvable with methods available. All variables are positive ($>0$).

Is there anyway to simplify this equation? Only real values are of interest.

Answer 1

The answer might be that there is no solution for large values of $t$ if you want the functions value to be 25,000. Here is a contour plot of the function for values of t and dl:

m = 1.000001; a = 0.566837276858827; q1 = 35931.22171;
ContourPlot[(E^(dl + a dl^(-1 + m) Gamma[1 - m, dl] - a dl^(-1 + m) Gamma[1 - m, dl t]) q1)/a,
 {dl, 0, 2}, {t, 0, 0.4},
 Contours -> {20000, 25000, 30000}, ContourLabels -> True, 
 ContourShading -> None,
 Frame -> True, FrameLabel -> (Style[#, Bold, 18] &) /@ {"dl", "t"},
 PlotRange -> {All, {0, 0.4}}, PlotRangePadding -> {{0, 0}, {0, 0}},
 PlotRangeClipping -> False, PlotPoints -> 100]

Once $t>0.2$, the values of the function are well over 25,000.

Answer 2

Another way to look at it. If dl and t are both positive and real, then Gamma[1 - m, dl t] essentially goes to 0 as t -> ∞ as shown by plotting.

m = 1000001/1000000;
a = 566837276858827/1000000000000000;
q1 = 3593122171/100000;

I use exact values so MMa doesn't create small imaginary results which messes up the plots.

Plot[Gamma[1 - m, dlt], {dlt, 0, 20}]

That term damps out at about dl*t = 15, so for large t we simplify your equation by throwing out the portion containing dl*t

Moving everything to one side we can form the function

fn[dl_]:= (E^(dl + a*dl^(-1 + m)*Gamma[1 - m, dl])*q1)/a - 25000

A plot shows that there are no positive roots for the function.

Plot[N[fn[dl]], {dl, .01, 2}, PlotRange -> {0, 200000}, WorkingPrecision -> 20]