# Complex infinity with integration

In trying to compute following integral (spherical coordinates): $$\int[\rho+p_r]4\pi r^2dr$$ With:$$\rho=\frac{1}{2} f_0 \left(\frac{\sqrt{b_0 r}}{r^2 \sqrt{r \sqrt{b_0 r}}}\right){}^n$$ $$p_r=\frac{f_0 \left(\frac{\sqrt{b_0 r}}{r^2 \sqrt{r \sqrt{b_0 r}}}\right){}^n \left(-2 n \sqrt{r \sqrt{b_0 r}}+n r-r\right)}{2 \left(\sqrt{b_0 r}-r\right)}$$ Wolfram code for this variables:

\[Rho] = 1/
2 f0 (Sqrt[r Subscript[b, 0]]/(
r^2 Sqrt[r Sqrt[r Subscript[b, 0]]]))^n
pr = (f0 (Sqrt[r Subscript[b, 0]]/(
r^2 Sqrt[r Sqrt[r Subscript[b, 0]]]))^
n (-r + n r - 2 n Sqrt[r Sqrt[r Subscript[b, 0]]]))/(
2 (-r + Sqrt[r Subscript[b, 0]]))


Where $$f_0$$ and $$b_0$$ is constants ($$b_0>0$$, $$f_0\in{\rm I\!R}$$). When I'm trying to compute the integral from first equation, I'm getting something like that: $$\Phi = -\Bigg[8 \pi \text{f0} \left(\frac{\sqrt{r \sqrt{b_0 r}}}{r^3}\right){}^n \left(\frac{b_0^3 r^3 \, _2F_1\left(1,6-\frac{9 n}{2};7-\frac{9 n}{2};\frac{r}{\sqrt{r b_0}}\right)}{3 n-4}-\frac{6 b_0^2 n r \left(r \sqrt{b_0 r}\right){}^{3/2} \, _2F_1\left(1,\frac{1}{2} (13-9 n);\frac{3}{2} (5-3 n);\frac{r}{\sqrt{r b_0}}\right)}{9 n-13}+\frac{3 (n-2) r \left(b_0 r\right){}^{5/2} \, _2F_1\left(1,7-\frac{9 n}{2};8-\frac{9 n}{2};\frac{r}{\sqrt{r b_0}}\right)}{9 n-14}\right)\bigg] \Bigg/\Bigg[3 b_0^3\Bigg]$$ Wolfram code:

Integrate[(\[Rho] + pr)*4 \[Pi]*r^2, r]

-((8 f0 \[Pi] (Sqrt[r Sqrt[r Subscript[b, 0]]]/r^3)^
n ((r^3 Hypergeometric2F1[1, 6 - (9 n)/2, 7 - (9 n)/2, r/Sqrt[
r Subscript[b, 0]]]
\!$$\*SubsuperscriptBox[\(b$$, $$0$$, $$3$$]\))/(-4 + 3 n) + (
3 (-2 + n) r Hypergeometric2F1[1, 7 - (9 n)/2, 8 - (9 n)/2, r/
Sqrt[r Subscript[b, 0]]] (r Subscript[b, 0])^(5/2))/(-14 +
9 n) - (6 n r Hypergeometric2F1[1, 1/2 (13 - 9 n), 3/2 (5 - 3 n),
r/Sqrt[r Subscript[b, 0]]]
\!$$\*SubsuperscriptBox[\(b$$, $$0$$, $$2$$]\) (r Sqrt[
r Subscript[b, 0]])^(3/2))/(-13 + 9 n)))/(3
\!$$\*SubsuperscriptBox[\(b$$, $$0$$, $$3$$]\)))


But when I'm trying to build plot with arbitrary values of parameters $$f_0$$, $$b_0$$, $$r$$, output gives my ComplexInfinity. What should I do to avoid hypergeometric function, which gives me this ComplexInfinity? Any help will be appreciated. Best regards, Alex.

• There is an unremovable singularity in \[Rho]+pr caused by the denominator 2 (-r + Sqrt[r Subscript[b, 0]]). May 5, 2021 at 17:46

With a command like

N[TableForm[Table[{n, x,
Hypergeometric2F1[1, (1/2)*(13 - 9*n), (3/2)*(5 - 3*n), x],
Hypergeometric2F1[1, 6 - (9*n)/2, 7 - (9*n)/2, x]}, {n, 1,
5}, {x, 0, 1.3, 0.13}]]]


you see that starting from n (integer) > 1 the Hypergeo functions show infinity the one or the other. For argument (sqrt[r/b0]) > 1 the numerical values get complex. You only can do a plot of the potential like

f0 = 1; b0 = 1; n = 1; Plot[(-(1/(3*(-4 + 3*n)*(-13 + 9*n))))*8*f0*Pi*
r^(3 - (9*n)/4)*(6*(4 - 3*n)*n*r^(1/4)*
Hypergeometric2F1[1, (1/2)*(13 - 9*n), (3/2)*(5 - 3*n), Sqrt[r]/Sqrt[b0]] +
(-13 + 9*n)*(2 - n + (-1 + n)*
Hypergeometric2F1[1, 6 - (9*n)/2, 7 - (9*n)/2,
Sqrt[r]/Sqrt[b0]])*b0^(1/4))*b0^(-(1/4) + n/4), {r, 0, b0}]