# How to maximize the modulus of a multivariate complex-valued function?

My problem is the following.

I have a messy, complex-valued function

 H[L_, G_, l_, g_] = -6.2593722277693634*10^(-9)*(4.303126009293465*10^(-11)*
Sqrt[1 - G^2/L^2]
L^4 (1/2 + 2.420508380227574*^-11 L^4) Cos[
l] + (1.058540218602996*10^(-16)* L^6 +
2.8468949665960885*10^(-27)* L^10) Cos[g + l] -
4.303126009293465*10^(-11)* Sqrt[1 - G^2/L^2]
L^4 (9/4 + 5.378907511616831*10^(-11)*L^4) Cos[
2 g + l] + (3.2273445069700985*10^(-11)* L^4 +
5.786529203705594*10^(-22)*L^8) Cos[2 g + 2 l] +
3.2273445069700985*10^(-11)* Sqrt[1 - G^2/L^2]
L^4 Cos[
2 g + 3 l] + (1.76423369767166*10^(-16)*L^6 +
3.3213774610287704*10^(-27) *L^10) Cos[3 g + 3 l] +
1.0126426106484789*10^(-21)* L^8 Cos[4 g + 4 l] +
5.978479429851786*10^(-27)* L^10 Cos[5 g + 5 l]) - 8.913267176762544*10^(-10)*(6.798114774925959*10^(-26)*
L^4 Cos[81.6814089933346 + 2 g + 2 l] +
1.2188810209227538*10^(-36)*
L^8 Cos[81.6814089933346 + 2 g + 2 l] +
2.1330417866148187*10^(-36)*
L^8 Cos[163.3628179866692 + 4 g + 4 l] +
2.4047395914526166*10^(-12)* Sqrt[1 - G^2/L^2]
L^4 (-Sin[l] + Sin[62.83185307179586 + l]) +
1.1641384666752184*10^(-22)*Sqrt[1 - G^2/L^2]
L^8 (-Sin[l] + Sin[62.83185307179586 + l]) +
9.100767021362843*10^(-18)*
L^6 (Sin[g + l] - Sin[81.6814089933346 + g + l]) +
2.447609204634166*10^(-28)*
L^10 (Sin[g + l] - Sin[81.6814089933346 + g + l]) +
5.055981678534912*10^(-18)*
L^6 (Sin[3 (g + l)] - Sin[3 (81.6814089933346 + g + l)]) +
9.518480240243979*10^(-29)*
L^10 (Sin[3 (g + l)] - Sin[3 (81.6814089933346 + g + l)]) +
1.0279958659463498*10^(-28)*
L^10 (Sin[5 (g + l)] - Sin[5 (81.6814089933346 + g + l)]) +
6.763330100960487*10^(-12)* Sqrt[1 - G^2/L^2]
L^4 (-Sin[2 g + l] + Sin[100.53096491487335 + 2 g + l]) +
1.6168589814933596*10^(-22)*Sqrt[1 - G^2/L^2]
L^8 (-Sin[2 g + l] + Sin[100.53096491487335 + 2 g + l]) -
1.0019748297719237*10^(-12)* Sqrt[1 - G^2/L^2]
L^4 (-Sin[2 g + 3 l] + Sin[226.19467105846508 + 2 g + 3 l]));


in four variables defined on a complex domain, Dom which is the cartesian product of

• Two complex balls Ball[La,rL] and Ball[Ga,rG] to which L and G belong respectively.
• Two open complex sets Reg1 and Reg2 to which l and g belong respectively.

I am trying to maximize the modulus of H with (L, G, l, g) varying over Dom.

The problems I have are

1. I don't know how to compute the cartesian product of the sets Ball[La, rL], Ball[Ga, rG], Reg1 and Reg1.

2. I would like to have an expression of the kind

Maximize[H, {L, G, l, g} ∈ Dom]


but I don't know if this is the right way to proceed.

Basically, I need to maximize the modulus of a complex-valued function of many complex variables on a specific domain and I have no idea of how to do that.

• Never use upper-case letters for functions as these will conflict with Mathematica's internal function names. For example D is the derivative operator and your code will never run in its present, un-syntactic form. – David G. Stork Jan 8 '18 at 23:40
• If this a question about the Mathematica software, please formulate it as expressions interpretable by Mathematica, and show us what you've tried as a solution. If not, you are asking the wrong group. – John Doty Jan 8 '18 at 23:46
• I've edited my post. Sorry for not being clear, it's my first time asking a question here. – Santiago Barbieri Jan 9 '18 at 0:01
• None of this makes any sense in Mathematica context. For example, Disk is a graphical object, not a set. Can you formulate this as a Mathematica problem? – John Doty Jan 9 '18 at 0:06
• Re-edited :) Hope it's more clear now. Just take the expression for the function H; I have no clue on how to maximize its modulus on its domain (which is the cartesian product of four complex sets). – Santiago Barbieri Jan 9 '18 at 0:55