I'm new to using mathematica, but I want to do a plot of this equation below:

$$ F(\nu) = \frac{4 \pi h \nu^{4}}{c^{2}} \int_{r_{i}}^{r_{o}} \frac{r d r}{\exp \left[\left(\frac{3 \pi}{2}\right)^{1 / 4} \frac{h \nu}{k T_{*}}\left(\frac{r}{R_{*}}\right)^{3 / 4}\right]-1} $$ where $R_* = 2.5(696.34*10^6)$, $r_i = 6*R_*$. $r_0 = 7*10^4*R_*$, $T=4000$, $h$ is de Planck constant, $c$ is speed of light and $k$ is the Bolztmann constant. The plot should be from $\nu_1 = 10^{10}Hz$ to $\nu_2 = 10^{16}Hz$. All values here are with the right units when using SI, so don't bother with it.

My main problem is that I do not know how can I set a function like this inside Mathematica because it is a function of $\nu$ and it has an integral in it. I know how to do integrals, but in this case, I have to compute the integral to each one of the $\nu$ values of my domain. I have no clue where to begin. Can someone help me?

Oh, it is supposed to look like this

EDIT: I followed the tip from the first comment but now I have another problem: The mathematica does not compute my integral. As it is indicate in the comment, I did:

fun[a_?NumericQ] :=  NIntegrate[((9.3*(10^(-50))) (a^4)*x)/(e^(6.89*(10^(-24))*a*x^(3/4)) - 1), {x,6*2.5*(696.34*(10^6)), 7*(10^4)*2.5*(696.34*(10^6))}]; Plot[fun[a], {a, 10^10, 10^16}]

but I get a black graph. It got me back a message saying:

The integrand has evaluated to non-numerical values for all sampling points in the region with boundaries

"Further output of NIntegrate::inumr will be suppressed during this calculation"

  • 1
    $\begingroup$ fun[a_?NumericQ] := NIntegrate[a*x, {x, 0, 1}] ; Plot[fun[a], {a, 1, 5}] $\endgroup$
    – I.M.
    May 7, 2021 at 2:42
  • $\begingroup$ Thanks! It was a valuable tip! $\endgroup$
    – kplt
    May 7, 2021 at 3:18
  • $\begingroup$ Post the formulas by Mathematica code. $\endgroup$
    – cvgmt
    May 7, 2021 at 6:36

1 Answer 1


R = 25 (69634*^3);
ri = 6 R;
r0 = 7*^4 R;
T = 4000;

h = QuantityMagnitude[Quantity[1, "PlanckConstant"] // UnitConvert];

c = QuantityMagnitude[Quantity[1, "SpeedOfLight"] // UnitConvert];

k = QuantityMagnitude[Quantity[1, 
       "BoltzmannConstant"] // UnitConvert];

F[v_?NumericQ] := 4 Pi h v^4/c^2 * NIntegrate[
  r/(Exp[(3 Pi/2)^(1/4) h v/(k T) (r/R)^(3/4)] - 1),
   {r, ri, r0}]

LogLogPlot[F[v], {v, 10^10, 10^16}]

enter image description here


Plot[Log10[F[10^v]], {v, 10, 16},
 Frame -> True,
 FrameLabel -> {"log v  (Hz)", "log F(v)"}]

enter image description here


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