How to find the minimum value of the real part of a complex function

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Question:How to find the minimum value of the real part of a complex function

Posted: ijuptilk 100 Product: Maple 18

April 05 2022

Please, how do I find the minimum of the real part of a complex function? I tried min ( ) function it didn't work.

Find attached the fileFinding_min_zero.mw

Import packages

>	restart: with(ArrayTools): with(Student:-Calculus1): with(LinearAlgebra): with(ListTools):with(RootFinding):with(ListTools):

Parameters

>	gamma1 := .1093: alpha3 := -0.1104e-2: k[1] := 610^(-12): d:= 0.2e-3: xi:= -0.01: theta0:= 0.1e-3: eta[1]:= 0.240e-1: alpha:= 1-alpha3^2/(gamma1eta[1]): c:= alpha3xialpha/(eta[1](4k[1]q^2/d^2-alpha3xi/eta[1])): theta_init:= theta0sin(Piz/d): n:= 10:

Assign g for q and plot g

>	g := q-(1-alpha)tan(q)-ctan(q): plot(g, q = 0 .. 3*Pi, view = [DEFAULT, -30.. 10]);

Set q as a complex

Assume and subsitute the result into g and equate the real and complex part to zero, and solve for x and y.

>	f := subs(q = x+I*y, g): b1 := evalc(Re(f)) = 0: b2 := evalc(Im(f)) = 0:

Compute the Special Asymptotes

This asymptote is coming from the from the definition of

>	qstar := (fsolve(1/c = 0, q = 0 .. infinity)):

Compute Odd asymptote

First, Since , then an asymptote occurs at In general, we have
"q= ((2 k+1)Pi)/(2). "
Next, we compute the entry of the Oddasymptotes that is close to qstar (special asymptote) as assign it to
ModifiedOaddAsym, and then find the minimum of the ModifiedOaddAsym. Searchall Function returns

the index of an entry in a list.

>	OddAsymptotes := Vector[row]([seq(evalf((1/2(2j+1))*Pi), j = 0 .. n)]); ModifiedOddAsym := abs(`~`[`-`](OddAsymptotes, qstar)); qstarTemporary := min(ModifiedOddAsym); indexOfqstar2 := SearchAll(qstarTemporary, ModifiedOddAsym); qstar2 := OddAsymptotes(indexOfqstar2);

$OddAsymptotes := Vector(4, {(1) = ` 1 .. 11 `*Vector[row], (2) = `Data Type: `*anything, (3) = `Storage: `*rectangular, (4) = `Order: `*Fortran_order})$

$ModifiedOddAsym := Vector(4, {(1) = ` 1 .. 11 `*Vector[row], (2) = `Data Type: `*anything, (3) = `Storage: `*rectangular, (4) = `Order: `*Fortran_order})$

(4.2.1)

Compute x and y

Here, we solve for and within the min. and max. of qstar2 and qstar, and substitute the results into .

AreThereComplexRoots := type(true, 'truefalse');
try
   soln1:= fsolve({b1, b2}, {x = min(qstar2, qstar) .. max(qstar2, qstar), y = 0 .. infinity});
   soln2:= fsolve({b1, b2}, {x = min(qstar2, qstar) .. max(qstar2, qstar), y = -infinity .. 0});
   qcomplex1 := subs(soln1, x+I*y);
   qcomplex2 := subs(soln2, x+I*y);
catch:
   AreThereComplexRoots := type(FAIL, 'truefalse');
end try;

(4.3.1)

Compute the rest of the Roots

In this section we compute the roots between each asymptotes.

OddAsymptotes := Vector[row]([seq(evalf((1/2)*(2*j+1)*Pi), j = 0 .. n)]);
AllAsymptotes := sort(Vector[row]([OddAsymptotes, qstar]));
if AreThereComplexRoots then
gg := [qcomplex1, qcomplex2, op(Roots(g, q = 0.1e-3 .. AllAsymptotes[1])), seq(op(Roots(g, q = AllAsymptotes[i] .. AllAsymptotes[i+1])), i = 1 .. n)];
elif not AreThereComplexRoots then
gg := [op(Roots(g, q = 0.1e-3 .. AllAsymptotes[1])), seq(op(Roots(g, q = AllAsymptotes[i] .. AllAsymptotes[i+1])), i = 1 .. n)];
end if:

$OddAsymptotes := Vector(4, {(1) = ` 1 .. 11 `*Vector[row], (2) = `Data Type: `*anything, (3) = `Storage: `*rectangular, (4) = `Order: `*Fortran_order})$

$AllAsymptotes := Vector(4, {(1) = ` 1 .. 12 `*Vector[row], (2) = `Data Type: `*anything, (3) = `Storage: `*rectangular, (4) = `Order: `*Fortran_order})$