# I failed to evaluate double integral

I try to evaluate this symbolic integral and evaluate its two series expansions according to certain variables, the plot the output providing some numerical values.

This is a relativistic rotational hollow disk. This one my thought experiments I try to figure it out. I am a hobbyist physicist

f1 = (y (-z +
y Cos[x]))/(((1 - (\[Omega] y)^2/h^2)^(1/2)) ((z^2 + y^2 -
2 z y Cos[x])^(3/2)));
f2 = Assuming[0 < z < 0.99 ymin  && 0.99 h > \[Omega] y,
Integrate[f1, {y, ymin, ymax}, {x, 0, 2 Pi}] // FullSimplify];
S2 = Assuming[0 < z < 0.99 ymin  && 0.99 h > \[Omega] y,
Series[f2, {z, 0, 3}, {\[Omega], 0, 4}] // FullSimplify]
S3 = Assuming[0 < z < 0.99 ymin  && 0.99 h > \[Omega] y,
Series[f2, {w, 0, 3}] // FullSimplify]
\[Omega] = 1;
w = z \[Omega]^2;
h = 1000;
ymin = 500;
ymax = 985;
Plot[f2, {w, 0, 495}]


My Question is: I takes very very long time, and I lost the hope to see result. I use the trial version of the most recent Mathematica.

Is there something wrong in this code?

------------ at Aug 28 2020 --------------

What if I try to fix z and plot versus omega provided that ymin>z>ymax. There will be divergence at z, so I tried to divide the integral to two integrals and take the limit at z to solve the improper integrations, the one that has ymin1>z>ymax1, and the other that has ymin2>z>ymax2. It didn't give me any result. Here is a trial to fix z and do series and plot with respect to omega.

f1[z_?NumericQ, \[Omega]_?NumericQ, h_?NumericQ, ymin1_?NumericQ,
ymax1_?NumericQ] :=
NIntegrate[(y (-z +
y Cos[x]))/(((1 - (\[Omega] y)^2/h^2)^(1/2)) ((z^2 + y^2 -
2 z y Cos[x])^(3/2))), {y, ymin1, ymax1}, {x, 0, 2 Pi},
f2[z_?NumericQ, \[Omega]_?NumericQ, h_?NumericQ, ymin2_?NumericQ,
ymax2_?NumericQ] :=
NIntegrate[(y (-z +
y Cos[x]))/(((1 - (\[Omega] y)^2/h^2)^(1/2)) ((z^2 + y^2 -
2 z y Cos[x])^(3/2))), {y, ymin2, ymax2}, {x, 0, 2 Pi},
f3 = f1 + f2;
L1 = Limit[f1[1, \[Omega], 1000, 0, ymx1], ymx1 -> 1,
Assumptions -> ymx1 < 1]
L2 = Limit[
f2[1, \[Omega], 1000, ymn2, ymx2], {ymn2 -> 1, ymx2 -> 250},
Assumptions -> ymn2 > 1 && ymx2 < 250  && \[Omega] < 4]
L3 = L1 + L2

S1 = Series[L1, {\[Omega], 0, 4}]
S2 = Series[L2, {\[Omega], 0, 4}]
S3 = Series[(L1 + L2), {\[Omega], 0, 4}]

P1 = Plot[L1, {\[Omega], 0, 4}, AxesLabel -> {"\[Omega]", "L1"}]
P2 = Plot[L2, {\[Omega], 0, 4}, AxesLabel -> {"\[Omega]", "L2"}]
P3 = Plot[(L1 + L2), {\[Omega], 0, 4},
AxesLabel -> {"\[Omega]", "L3"}]


And what if I need to get series and plot w.r.t. z also?

• Also your plot range makes no sense because w = z \[Omega]^2 and f2 will have a mixture of y and z. Plot needs a single independent variable. Aug 19 '20 at 9:59
• Hi flinty, 1- what is Rubi; role? 2- y should not be an independent variable after integration from ymin to ymax Aug 19 '20 at 10:17
• Apologies - I was integrating wrt x. Aug 19 '20 at 10:20
• Rubi is a package for rule-based-integration which can sometimes solve integrals when Mathematica cannot. It appears that even Rubi cannot do it when integrating over both y and x. You should therefore look into numerically integrating it. Aug 19 '20 at 10:23
• @flinty Is numerical integration same but NIntegrate instead of Integrate? Aug 19 '20 at 10:34

It looks like you'll have to go for numerical integration as neither Mathematica nor Rubi could calculate a symbolic integral.

f2[z_?NumericQ, ω_?NumericQ, h_?NumericQ, ymin_?NumericQ, ymax_?NumericQ] :=
NIntegrate[
(y (-z + y Cos[x]))/(((1 - (ω y)^2/h^2)^(1/2)) ((z^2 + y^2 - 2 z y Cos[x])^(3/2)))
, {y, ymin, ymax}, {x, 0, 2 Pi}, Method -> "LocalAdaptive"]

Plot[f2[z, 1.0, 1000.0, 500.0, 985.0], {z, 0, 495},
AxesLabel -> {"z", "f2"}]


• Great flinty, How much time it took from you to graph? And which version of Mathematica? Aug 19 '20 at 11:25
• @AhmedKamalKassem with Method -> "LocalAdaptive"` it's much faster, around 10s, but without that it takes a long time > 1min. I'm using Windows 10 Mathematica v12.1.1.0 home. Aug 19 '20 at 11:44
• I tried the double integral, bot this time from 0 to ymax, so there is a divergence at y=z, if ymax>z>ymin, so, to avoid divergence for the improper integration I took the limit, but I don't get a result: the new code: Aug 27 '20 at 12:47
• @fliny What if I try to fix z and plot versus omega, but z has the condition ymin>z>ymax. There will be divergence at z, so I tried to divide the integral to two integrals and take the limit at z to solve the improper integrations, the one that has ymin1>z>ymax1, and the other that has ymin2>z>ymax2. It didn't give me any result. I tried to put the code here in the comment it exceeded the number of letters allowed. Aug 27 '20 at 13:28