de-CH
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math math-format graphie
Advanced Operations with Complex Numbers
bsp-komplex-01
custom
13552
randRangeExclude(-3,3,[0]) randRangeExclude(-3,3,[0]) randRangeExclude(-3,3,[0]) randRangeExclude(-4,4,[0]) (-Y)*B + X*A X*B + A*Y sqrt(re0*re0 + im0*im0) (2*3.1415 + atan2(im0, re0)) % (2*3.1415) max(0,r0 - 2) r0 + 2 (2*3.1415 + t0 - 3.1415/14) % (2*3.1415) (t0 + 3.1415/14) % (2*3.1415)

Let {\color{blue}z} = X + Yi and {\color{green}w} = A + {\color{red}B} \cdot i be two complex numbers.

Determine {\color{red}B}, such that z \cdot w lies in the region D (piece of an annulus).

graphInit({ range: [[-20, 20], [-20, 20]], scale: 11, gridStep: [2,2], tickStep: 1, labelStep: 2, axisArrows: "->" }); label([18,0], "\\operatorname{Re}", "above right"); label([0,18], "\\operatorname{Im}", "above right"); // z circle([X, Y], 0.35, { fill: "blue" }); // sektor var p11 = [rmin * Math.cos(tmin), rmin * Math.sin(tmin)]; var p12 = [rmin * Math.cos(tmax), rmin * Math.sin(tmax)]; var p21 = [rmax * Math.cos(tmin), rmax * Math.sin(tmin)]; var p22 = [rmax * Math.cos(tmax), rmax * Math.sin(tmax)]; line(p11, p21); line(p12, p22); arc([0,0], rmin, tmin * 180 / 3.1415, tmax * 180 / 3.1415); arc([0,0], rmax, tmin * 180 / 3.1415, tmax * 180 / 3.1415); label([re0, im0], "D");
{\color{red}B=}
[document.getElementById("custom_input").value]
var re = X*A - Y*guess; var im = X*guess + Y*A; var r = Math.sqrt(re*re + im*im); var t = (2*3.1415 + atan2(im, re)) % (2*3.1415); var dt = Math.max(Math.abs((t - tmin + 2*3.1415) % (2*3.1415)), Math.abs((t - tmax - 2*3.1415) % (2*3.1415))); return (r >= rmin) && (r <= rmax) && dt <= 3.1415/7;

The vertical grey line shows all values that {\color{green}w} can take for different values of {\color{red}B}.

line([A,-20], [A,20], {stroke: "grey"});

A strategy is to estimate how large the argument of {\color{green}w} must be to rotate {\color{blue}z} in the direction of D.

Then, we can choose an appropriate scaling by fixing the absolute value {\color{green}|w|}.

Alternatively, we consider the set of all complex numbers of the form {\color{blue}z} \cdot (A + {\color{red}B} \cdot i). This is a line in the complex plane.

plot(function(x){ return (X/-Y*x) + Y*A+X/Y*X*A; }, [-20,20], { stroke: "red" });

Now we choose a point in D and find the parameter {\color{red}B}, such that the point lies on the line.

circle([re0, im0], 0.35, { fill: "orange" });

A description of the line with the given numbers is \begin{pmatrix} x\\ y \end{pmatrix} = \begin{pmatrix} A*X \\ A*Y \end{pmatrix} + {\color{red}B} \begin{pmatrix} -Y \\ X \end{pmatrix} .

Substitute the coordinates of the point on the left-hand side and find {\color{red}B} as \begin{pmatrix}re0 \\ im0 \end{pmatrix} = \begin{pmatrix} A*X \\ A*Y \end{pmatrix} + {\color{red}B} \begin{pmatrix} -Y \\ X \end{pmatrix} .

circle([A, B], 0.35, { fill: "red" });

Our solution is {\color{red}B} = B, so {\color{green}w} = A {\color{red} + B} \cdot i.