# How do I find the zero of this function?

How do I find the zero of this function:

Ef=5.53;
e=1.6*10^-19;
qc[w1_?NumericQ] :=
w1 (1/(2 a[w1]) (1 + Sqrt[1 - 4/(27 a[w1])])^(1/3) +
1/(2 a[w1]) (1 - Sqrt[1 - 4/(27 a[w1])])^(1/3));

a[w1_?NumericQ] :=
Piecewise[{{2/3,
w1 < 0.62}, {0.9069 + 0.3577 w1^(-2/3) - 1.0565 w1^(-1) +
1.478 w1^(-4/3) - 0.4524 w1^(-5/3), w1 > 0.62}}];
wmax2[w1_?NumericQ, T_?NumericQ] := Min[T - Ef, T/2];
s4[w1_?NumericQ, T_?NumericQ] := (Pi e^4)/
T (Log[wmax2[w1, T]/qc[w1]] + T/(T - wmax2[w1, T]) - T/(
T - qc[w1]) + 2 Log[(T - wmax2[w1, T])/(T - qc[w1])]);


If I plot it it crosses the x axis, so there is a zero, but NSolve[s4[100,T]==0,T] gives a complex number as a solution and says to use Refine, and Refine can't find the zero.

• FindRoot perhaps? You haven't defined qc for us, so we can't investigate more. – John Doty Jul 4 '18 at 14:50
• Use FindRoot. Can't say more, because you didn't give full definitions of your symbols (e.g. wmax2, qc). You have asked 48 questions on this site so far, you should know how to ask a good question by now. – MarcoB Jul 4 '18 at 14:50
• @MarcoB added the rest of the function I am sorry. – mattiav27 Jul 4 '18 at 15:01
• @mattiav27 a is still undefined. The best way to check for these things is to copy back your posted code in a clean MMA session (see this Q) and try to execute it. It has to work under those conditions, because that's what we will be trying on our own computers. – MarcoB Jul 4 '18 at 15:26
• @MarcoB Added the definition of a – mattiav27 Jul 4 '18 at 16:53

With the full code, we can explore the plot to find an initial estimate of the root. COnsidering the extremely small values of the function, I used a LogPlot on its absolute value:

LogPlot[Abs@s4[100, T], {T, 100, 250}, PlotRange -> All]


We confirm that with a regular plot:

Plot[s4[100, T], {T, 120, 250}, PlotRange -> All]


A good estimate for the root appears to be around $180$. We can feed that into FindRoot as a starting point for root search:

FindRoot[s4[100, T], {T, 180}]
(* Out: {T -> 183.495} *)


With a large value for PlotPoints we can also use purely graphical approach to get the roots:

Normal[Plot[s4[100, T], {T, 120, 250}, PlotRange -> All,
MeshFunctions -> {#2 &}, Mesh -> {{0}},
MeshStyle -> PointSize[Large], PlotPoints -> 300]] /.
Point[x_] :> {Point[x], Text[Framed @ Style[x, 14], Offset[{50,-30}, x]]}


Use

Text[Framed@Style[root = T/. FindRoot[s4[100, T], {T, x[[1]]}, WorkingPrecision->20 ], 12]


in place of Text[...] above to get