Puzzle Piece

Calculus Level 3

Let α \alpha be a complex number whose real and imaginary parts are plotted on the horizontal and vertical Cartesian axes.

Consider the following quantity:

Q = 1 + 1 2 ∑ k = 1 k = 7 α k 1 − 1 2 ∑ k = 1 k = 7 α k {\large Q} = \frac{\displaystyle 1 + \frac{1}{2} \sum_{k=1}^{k=7} \alpha^k}{\displaystyle 1 - \frac{1}{2} \sum_{k=1}^{k=7} \alpha^k}

Within the square region defined by ( − 2 < ℜ ( α ) < 2 ) (-2 < \Re(\alpha) < 2) and ( − 2 < ℑ ( α ) < 2 ) (-2 < \Im(\alpha) < 2) , what is the area of the sub-region for which ∣ Q ∣ < 1 |Q| < 1 ? This is the blue portion of the diagram below.


The answer is 9.077.

Steven Chase by Steven Chase , 37, USA

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1 solution

Karan Chatrath
Dec 10, 2019

Simulation code attached below. Answer is: A S = 9.0822 \boxed{AS=9.0822} . 

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clear all
clc

dx = 0.001;
dy = 0.001;
AS = 0;

for x = -2:dx:2
    for y = -2:dy:2
        % Computing alpha denoted as A:
        A = x + j*y;

        % Computing Q:
        S = A^1 + A^2 + A^3 + A^4 + A^5 + A^6 + A^7;
        Q = (1 + 0.5*S)/(1 - 0.5*S);

        % Condition check:
        if abs(Q)<1
            % Area computation:
            AS = AS + dx*dy;
        end
    end
end

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