# Multivariate integration of a compicated expression

I have an expression in multiple variables that is something like

4.85746*10^-7 Cos[ϕ] (1 +
1/2 (Abs[(-1.5782 Sqrt[1 - z^2 Sin[ϕ]^2] +
1.329 Sqrt[1 - 0.70913 z^2 Sin[ϕ]^2])/(1.5782 Sqrt[
1 - z^2 Sin[ϕ]^2] +
1.329 Sqrt[1 - 0.70913 z^2 Sin[ϕ]^2])]^2 +
Abs[(1.329 Sqrt[1 - z^2 Sin[ϕ]^2] -
1.5782 Sqrt[1 - 0.70913 z^2 Sin[ϕ]^2])/(1.329 Sqrt[
1 - z^2 Sin[ϕ]^2] +
1.5782 Sqrt[1 - 0.70913 z^2 Sin[ϕ]^2])]^2) Cos[
2 ArcSin[
z Sin[ϕ]]] - ((1 +
1/2 (-Abs[(-1.5782 Sqrt[1 - z^2 Sin[ϕ]^2] +
1.329 Sqrt[
1 - 0.70913 z^2 Sin[ϕ]^2])/(1.5782 Sqrt[
1 - z^2 Sin[ϕ]^2] +
1.329 Sqrt[1 - 0.70913 z^2 Sin[ϕ]^2])]^2 -
Abs[(1.329 Sqrt[1 - z^2 Sin[ϕ]^2] -
1.5782 Sqrt[
1 - 0.70913 z^2 Sin[ϕ]^2])/(1.329 Sqrt[
1 - z^2 Sin[ϕ]^2] +
1.5782 Sqrt[
1 - 0.70913 z^2 Sin[ϕ]^2])]^2))^2 (Cos[
2 ArcSin[0.842099 z Sin[ϕ]] -
2 ArcSin[z Sin[ϕ]]] +
1/2 (Abs[(-1.5782 Sqrt[1 - z^2 Sin[ϕ]^2] +
1.329 Sqrt[
1 - 0.70913 z^2 Sin[ϕ]^2])/(1.5782 Sqrt[
1 - z^2 Sin[ϕ]^2] +
1.329 Sqrt[1 - 0.70913 z^2 Sin[ϕ]^2])]^2 +
Abs[(1.329 Sqrt[1 - z^2 Sin[ϕ]^2] -
1.5782 Sqrt[
1 - 0.70913 z^2 Sin[ϕ]^2])/(1.329 Sqrt[
1 - z^2 Sin[ϕ]^2] +
1.5782 Sqrt[1 - 0.70913 z^2 Sin[ϕ]^2])]^2) Cos[
2 ArcSin[z Sin[ϕ]]]))/(1 +
1/4 (Abs[(-1.5782 Sqrt[1 - z^2 Sin[ϕ]^2] +
1.329 Sqrt[1 - 0.70913 z^2 Sin[ϕ]^2])/(1.5782 Sqrt[
1 - z^2 Sin[ϕ]^2] +
1.329 Sqrt[1 - 0.70913 z^2 Sin[ϕ]^2])]^2 +
Abs[(1.329 Sqrt[1 - z^2 Sin[ϕ]^2] -
1.5782 Sqrt[1 - 0.70913 z^2 Sin[ϕ]^2])/(1.329 Sqrt[
1 - z^2 Sin[ϕ]^2] +
1.5782 Sqrt[
1 - 0.70913 z^2 Sin[ϕ]^2])]^2)^2 + (Abs[(-1.5782 \
Sqrt[1 - z^2 Sin[ϕ]^2] +
1.329 Sqrt[1 - 0.70913 z^2 Sin[ϕ]^2])/(1.5782 Sqrt[
1 - z^2 Sin[ϕ]^2] +
1.329 Sqrt[1 - 0.70913 z^2 Sin[ϕ]^2])]^2 +
Abs[(1.329 Sqrt[1 - z^2 Sin[ϕ]^2] -
1.5782 Sqrt[1 - 0.70913 z^2 Sin[ϕ]^2])/(1.329 Sqrt[
1 - z^2 Sin[ϕ]^2] +
1.5782 Sqrt[1 - 0.70913 z^2 Sin[ϕ]^2])]^2) Cos[
2 ArcSin[0.842099 z Sin[ϕ]]]))


I want to integrate this expression first w.r.t. ϕ with limits 0 to 7 π/18 and then indefinite integral Integrate[F, z] w.r.t z. I tried it in many ways, but it is very difficult for me to solve this integral. Can anyone help to find out the solution of this integral. I would be highly obliged.

• I highly doubt you can find an indefinite integral here. Such complicated functions are usually best solved numerically. – Feyre Jul 1 '16 at 11:08
• Are you certain that you need a symbolic result? What do you want to use it for? – Szabolcs Jul 1 '16 at 11:31
• after this integral i want to plot graph between thisfunction and z. – Anita Maheshwari Jul 2 '16 at 13:55

The integrand cannot be solved when $|z|\rightarrow1$ Therefore, let's integrate the function on a possible domain, numerically.

zdat=Table[NIntegrate[f, {Phi, 0, 7 Pi/18}, {z, 0, i}], {i, -0.95, 0.95, 0.1}];


This gives us a list of values, from which we can approximate a function for this part of the domain for z. Plotting this list gives us:

lp = ListPlot[data = Transpose[{Table[i, {i, -0.95, 0.95, 0.1}], zdat}]]


We can use this to formulate an approximate function for z. The function appears to be odd, and includes the point {0,0}, and we can try an odd polynomial.

nlm = NonlinearModelFit[data, a z + b z^3, {a, b}, z]//Normal


1.2737*10^-8 z + 8.36705*10^-9 z^3

Show[Plot[nlm, {z, -0.95, 0.95}, PlotStyle -> Red], lp]


Adding a fifth order term barely changes the plot.

nlm2 = NonlinearModelFit[data, a z + b z^3 + c z^5, {a, b, c}, z] //
Normal

Plot[{nlm, nlm2}, {z, -0.95, 0.95}, PlotStyle -> {Red, {Dashed, Black}}]


Therefore the third order polynomial appears to approximate the function you seek quite well.

$$1.2737\times 10^{-8} z + 8.36705\times 10^{-9} z^3$$

When we extend the domain, the integrand becomes highly oscillatory, the errors in the numerical integration become quite large.

At this point, the function becomes rather more complicated, the function is then better approximated with an interpolating function:

zdat = Table[NIntegrate[f, {\[Phi], 0, 7 Pi/18}, {z, 0, i},
Method -> {"GlobalAdaptive",MaxErrorIncreases -> 10000}], {i, -2.01, 2.01, 0.1}];

data = Re[zdat];
data = Transpose[{Table[i, {i, -2.01, 2.01, 0.1}], data}];
ip = Interpolation[data];
Show[Plot[ip[z], {z, -2.01, 1.9900000000000002}, PlotStyle -> Red],ListPlot[data]]


• but i want to plot in -2 to 2 range.Is their any possible solution for that . – Anita Maheshwari Jul 3 '16 at 10:27
• @AnitaMaheshwari I've extended my answer. – Feyre Jul 3 '16 at 12:28
• I am running these command again and again but it is showing me some error like its not a valid command for z and so, – Anita Maheshwari Jul 4 '16 at 4:15
• I have one more question this is the plot only for phi integral.their is any possiblity to integrate this function w.r.t z?? – Anita Maheshwari Jul 4 '16 at 4:16
• whatever you did may be its correct but the nature of the curve is not correct because the way you did in this is you are integrating w.r.t particular value every time that's why it will give you the nature of curve same as after integrating phi if you plot after phi integration is same as this curve.That's why i want separate indefinite integral of z. – Anita Maheshwari Jul 4 '16 at 9:48

One can use NDSolve to construct a particular "indefinite" integral, e.g., $$f(z) = \int_0^z f'(ζ) \; dζ$$

First define integrand to be the OP's expression. The the following computes its integral.

Clear[df];
df[z0_?NumericQ] :=
df[z0] = NIntegrate[integrand /. z -> z0, {ϕ, 0, 7 Pi/18},
Method -> {"GaussKronrodRule", "Points" -> 11},
MaxRecursion -> 20];
{fsol} = NDSolve[{f'[z] == df[z], f[0] == 0},
f, {z, -2., 2.}]; // AbsoluteTiming
(*  {1.89007, Null}  *)


The function is an InterpolatingFunction that can be obtained with f /. fsol or plotted with f[z] /. fsol. See the documentation for NDSolve for more examples.

Plot[f[z] /. fsol, {z, -2., 2.}]


• when I run this integral in my mathematica 10.0 it showed me many error .Can you please tell me what should i do to remove all these error? – Anita Maheshwari Jul 4 '16 at 4:27
• @AnitaMaheshwari Did you define integrand to be your integrand? – Michael E2 Jul 4 '16 at 5:08
• whatever you did may be its correct but the nature of the curve is not correct because the way you did in this is you are integrating w.r.t particular value every time that's why it will give you the nature of curve same as after integrating phi if you plot after phi integration is same as this curve.That's why i want separate indefinite integral of z. – Anita Maheshwari Jul 4 '16 at 9:47
• @AnitaMaheshwari Like I said from the get-go, integrating this function symbolically is impossible. An interpolating function, is as good as it gets. – Feyre Jul 4 '16 at 11:02
• can you please explain your code briefly because i tried that thing in mathematica 10 and i found nothing.Its a humble request if you can do this??taking main function to f. – Anita Maheshwari Jul 6 '16 at 4:18